Martin Goldstern, Jakob Kellner, Saharon Shelah and Wolfgang Wohofsky- Borel Conjecture and Dual...

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BOREL CONJECTURE AND DUAL BOREL CONJECTURE1

MARTIN GOLDSTERN, JAKOB KELLNER, SAHARON SHELAH, AND WOLFGANG WOHOFSKY2

A. We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simulta-neously.

I3

History. A set X of reals1 is called “strong measure zero” (smz), if for all functionsf : ω → ω there are4

intervalsIn of measure≤ 1/ f (n) coveringX. Obviously, a smz set is a null set (i.e., has Lebesgue measure5

zero), and it is easy to see that the family of smz sets forms aσ-ideal and that perfect sets (and therefore6

uncountable Borel or analytic sets) are not smz.7

At the beginning of the 20th century, Borel [Bor19, p. 123] conjectured:

Every smz set is countable.

This statement is known as the “Borel Conjecture” (BC). In the 1970s it was proved that BC isindependent,8

i.e., neither provable nor refutable.9

Let us very briefly comment on the notion of independence: A sentenceϕ is called independent of a set10

T of axioms, if neitherϕ nor¬ϕ follows fromT. (As a trivial example, (∀x)(∀y)x · y = y · x is independent11

from the group axioms.) The set theoretic (first order) axiomsystem ZFC (Zermelo Fraenkel with the axiom12

of choice) is considered to be the standard axiomatization of all of mathematics: A mathematical proof is13

generally accepted as valid iff it can be formalized in ZFC. Therefore we just say “ϕ is independent” ifϕ14

is independent of ZFC. Several mathematical statements areindependent, the earliest and most prominent15

example is Hilbert’s first problem, the Continuum Hypothesis (CH).16

BC is independent as well: Sierpinski [Sie28] showed that CH implies¬BC (and, since Godel showed17

the consistency of CH, this gives us the consistency of¬BC). Using the method of forcing, Laver [Lav76]18

showed that BC is consistent.19

Galvin, Mycielski and Solovay [GMS73] proved the followingconjecture of Prikry:

X ⊆ 2ω is smz if every comeager (denseGδ) set contains a translate ofX.

Prikry also defined the following dual notion:

X ⊆ 2ω is called “strongly meager” (sm) if every set of Lebesgue measure 1 containsa translate ofX.

The dual Borel Conjecture (dBC) states:

Every sm set is countable.

Prikry noted that CH implies¬dBC and conjectured dBC to be consistent (and therefore independent),20

which was later proved by Carlson [Car93].21

Numerous additional results regarding BC and dBC have been proved: The consistency of variants of22

BC or of dBC, the consistency of BC or dBC together with certain assumptions on cardinal characteristics,23

Date: 2011-05-28.2000Mathematics Subject Classification.Primary 03E35; secondary 03E17, 28E15.We gratefully acknowledge the following partial support: US National Science Foundation Grant No. 0600940 (all authors);

US-Israel Binational Science Foundation grant 2006108 (third author); FWF Austrian Science Fund grant P21651 and EU FP7 MarieCurie grant PERG02-GA-2207-224747 (second and fourth author); FWF grant P21968 (first and fourth author);OAW Doc fellowship(fourth author). This is publication 969 of the third author.

1In this paper, we use 2ω as the set of reals. (ω = 0,1, 2, . . ..) By well-known results both the definition and the theorem alsoworks for the unit interval [0, 1] or the torusR/Z. Occasionally we also write “x is a real” for “x ∈ ωω”.

1

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2 MARTIN GOLDSTERN, JAKOB KELLNER, SAHARON SHELAH, AND WOLFGANG WOHOFSKY

etc. See [BJ95, Ch. 8] for several of these results. In this paper, we prove the consistency (and therefore1

independence) of BC+dBC (i.e., consistently BC and dBC hold simultaneously).2

The problem. The obvious first attempt to force BC+dBC is to somehow combine Laver’s and Carlson’s3

constructions. However, there are strong obstacles:4

Laver’s construction is a countable support iteration of Laver forcing. The crucial points are:5

• Adding Laver real makes every old uncountable setX non-smz.6

• And this setX remains non-smz after another forcingP, provided thatP has the “Laver property”.7

So we can start with CH and use a countable support iteration of Laver forcing of lengthω2. In the final8

model, every setX of reals of sizeℵ1 already appeared at some stageα < ω2 of the iteration; the next Laver9

real makesX non-smz, and the rest of the iteration (as it is a countable support iteration of proper forcings10

with the Laver property) has the Laver property, and thereforeX is still non-smz in the final model.11

Carlson’s construction on the other hand addsω2 many Cohen reals in a finite support iteration (or12

equivalently: finite support product). The crucial points are:13

• A Cohen real makes every old uncountable setX non-sm.14

• And this setX remains non-sm after another forcingP, provided thatP has precaliberℵ1.15

So we can start with CH, and use more or less the same argument as above: Assume thatX appears at16

α < ω2. Then the next Cohen makesX non-sm. It is enough to show thatX remains non-sm at all17

subsequent stagesβ < ω2. This is guaranteed by the fact that a finite support iteration of Cohen reals of18

sizeℵ1 has precaliberℵ1.19

So it is unclear how to combine the two proofs: A Cohen real makes all old sets smz, and it is easy20

to see that whenever we add Cohen reals cofinally often in an iteration of length, say,ω2, all sets of any21

intermediate extension will be smz, thus violating BC. So wehave to avoid Cohen reals,2 which also22

implies that we cannot use finite support limits in our iterations. So we have a problem even if we find a23

replacement for Cohen forcing in Carlson’s proof that makesall old uncountable setsX non-sm and that24

does not add Cohen reals: Since we cannot use finite support, it seems hopeless to get precaliberℵ1, an25

essential requirement to keepX non-sm.26

Note that it is theproofsof BC and dBC that are seemingly irreconcilable; this is not clear for the27

models. Of course Carlson’s model, i.e., the Cohen model, cannot satisfy BC, but it is not clear whether28

maybe already the Laver model could satisfy dBC. (It is even still open whether a single Laver forcing29

makes every old uncountable set non-sm.) Actually, Bartoszynski and Shelah [BS03] proved that the Laver30

model does satisfy the following weaker variant of dBC (notethat the continuum has sizeℵ2 in the Laver31

model):32

Every sm set has size less than the continuum.33

In any case, it turns out that onecanreconcile Laver’s and Carlson’s proof, by “mixing” them “generi-34

cally”, resulting in the following theorem:35

Theorem. If ZFC is consistent, then ZFC+BC+dBC is consistent.36

We give a rather informal overview of the proof in Section 1.37

Prerequisites. To understand anything of this paper, the reader38

• should have some experience with finite and countable support iteration, proper forcing,ℵ2-cc,39

σ-closed, etc.40

• should know what a quotient forcing is,41

• should have seen some preservation theorem for proper countable support iteration,42

• should have seen some tree forcings (such as Laver forcing).43

To understand everything, additionally the following is required:44

• The “case A” preservation theorem from [She98], more specifically we build on the proof of [Gol93]45

(or [GK06]).46

• In particular, some familiarity with the property “preservation of randoms” is recommended. We47

will use the fact that random and Laver forcing have this property.1

2An iteration that forces dBC without adding Cohen reals was given in [BS10], using non-Cohen oracle-cc.

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BOREL CONJECTURE AND DUAL BOREL CONJECTURE 3

• We make some claims about (a rather special case of) ord-transitive models in Section 4.A. The2

reader can either believe these claims, or check them himself (by some rather straightforward3

proofs), or look up the proofs (of more general settings) in [She04] or [Kel].4

From the theory of strong measure zero and strongly meager, we only need the following two results5

(which are essential for our proofs of BC and dBC, respectively):6

• Pawlikowski’s result from [Paw96a] (which we quote as Theorem 1.2), and7

• Theorem 8 of Bartoszynski and Shelah’s [BS10] (which we quote as Lemma 3.2).8

We do not need any other results of Bartoszynski and Shelah’s paper [BS10]; in particular we do not use the9

notion of non-Cohen oracle-cc. (Note that the third author claims that our construction is more or less the10

same as a non-Cohen oracle-cc construction introduced in [She06], and that the extended version presented11

in [She10] is even closer to our preparatory forcing.)12

The reader does not have to know the original proofs of Con(BC) and Con(dBC), by Laver and Carlson,13

respectively.14

Acknowledgment. We thank Tomek Bartoszynski for pointing out Pawlikowski’s result [Paw96a] to us,15

and for many interesting discussions on the topic.16

Notation. Stronger conditions in forcing notions are smaller, i.e.,q ≤ p means thatq is stronger thanp.17

Let P ⊆ Qbe forcing notions. (As usual, we abuse notation by not distinguishing between the underlying18

set and the quasiorder on it.)19

• For p1, p2 ∈ P we write p1 ⊥P p2 for “ p1 andp2 are incompatible”. Otherwise we writep1 ‖ p2.20

• q ≤∗ p means thatq forces thatp is in the generic filter.21

• “P is a subforcing of Q” means that the relation≤P is the restriction of≤Q to P.22

• “P is an incompatibility-preserving subforcing of Q” means thatP is a subforcing ofQ and that23

p1 ⊥P p2 iff p1 ⊥Q p2 for all p1, p2 ∈ P.24

Let additionallyM be a countable transitive3 model (of a sufficiently large subset of ZFC) containingP.25

• “P is an M-complete subforcing ofQ” (or: P ⋖M Q) means thatP is a subforcing ofQ and: if26

A ⊆ P is in M a maximal antichain, then it is a maximal antichain ofQ as well. (Or equivalently:27

P is an incompatibility preserving subforcing ofQ and every predense subset ofP in M is predense28

in Q.) Note that this means that everyQ-generic filterG overV induces aP-generic filter overM,29

namelyGM≔ G ∩ P (i.e., every maximal antichain ofP in M meetsG ∩ P in exactly one point).30

In particular, we can interpret aP-nameτ in M as aQ-name. More exactly, there is aQ-nameτ′31

such thatτ′[G] = τ[GM] for all Q-generic filtersG. We will usually just identifyτ andτ′.32

• Analogously, ifP ∈ M andi : P→ Q is a function, theni is called anM-complete embedding if it33

preserves≤ (or at least≤∗) and⊥ and moreover: IfA ∈ M is predense inP, theni[A] is predense34

in Q.35

Annotated contents.36

Section 1, p. 4: We give a rather informal overview of the whole construction and proof.37

Section 2, p. 9: We introduce the family of ultralaver forcing notions and prove some properties.38

Section 3, p. 19: We introduce the family of Janus forcing notions and prove some properties.39

Section 4, p. 23: We define ord-transitive models and mentionsome basic properties. We define the40

“almost finite” and “almost countable” support iteration over a model. We show that in many41

respects they behave like finite and countable support, respectively.42

Section 5, p. 36: We introduce the preparatory forcing notionRwhich adds a generic forcing iterationP.43

Section 6, p. 44: Putting everything together, we show thatR∗Pω2 forces BC+dBC, i.e., that an uncount-44

ableX is neither smz nor sm. We show this under the assumptionX ∈ V, and then introduce a45

factorization ofR ∗ P that this assumption does not result in loss of generality.1

3We will also use so-called ord-transitive models, as definedin Section 4.A.

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1. T 2

In this section, we give a rather informal overview of (and background for) the proof. The emphasis is3

on giving the reader some vague understanding (i.e., a warm fuzzy feeling), at the expense of correctness4

of the claims (we point out some of the most blatant lies).5

1.A. The general setup.We assume CH in the ground model. We use aσ-closedℵ2-cc preparatory6

forcingR, which adds a generic “alternating iteration” (as defined below) P = (Pα,Qα)α<ω2. Moreover,R7

forces thatP is ccc. The forcing notion to get BC+dBC is the compositionR ∗ Pω2.8

We say thatP is an “alternating iteration” ifP = (Pα,Qα)α<ω2 is a forcing iteration of lengthω2 satisfy-9

ing the following:10

• At every even stepα, (Pα forces that)Qα is an “ultralaver forcing” (described below).11

• At every odd stepα, (Pα forces that)Qα is a “Janus forcing” (described below).12

• However, instead of using either a Janus or an ultralaver forcing, we are at any step allowed just to13

“do nothing”, i.e., setQα = ∅.14

• At a limit stepδ, we take “partial countable support” limits. (This means more or less:Pδ is a15

subset of the countable support limit of (Pα)α<δ and contains⋃

α<δ Pα and has some other natural16

properties.)17

1.B. Ultralaver forcing. Let D = (Ds)s∈ω<ω be a system of ultrafilters. The “ultralaver forcing”LD con-18

sists of treesp with the following property: For every nodes ∈ p above the stem the set of immediate19

successors ofs is in Ds. So this is aσ-centered variant of Laver forcing. Of course this forcing adds a20

naturally defined generic real, called ultralaver real.21

We will basically need two properties of ultralaver forcing: The first one is preservation of positivity:22

(1.1) LD preserves Lebesgue outer measure positivity of ground model sets.4

The second one is killing of smz sets:

For every uncountable setX in the ground model,LD forces thatX is non-smz.

Actually, we should formulate this claim in a stronger form.Let us first quote a result of Pawlikowski [Paw96a],23

which is essential for the part of our proof that shows BC:24

Theorem 1.2. X ⊆ 2ω is smz iff X + F is null for every closed null set F.25

Moreover, for every dense Gδ set H we canconstruct(in an absolute way) a closed null set F such that for26

every X⊆ 2ω with X+ F null there is t∈ 2ω with t+ X ⊆ H.27

So we can actually show the following:28

(1.3)We can construct from the ultralaver real in an absolute way a(code for a) closednull setF such thatX+F is (outer Lebesgue measure) positive for every uncountableground model setX.

It is an easy exercise to show that Theorem 1.2 implies the following fact.29

Fact 1.4. Assume thatP = (Pα,Qα : α < ω2) is an iteration with direct limitPω2 satisfying the following:30

• For cofinally manyα < ω2, Qα makes every old uncountable set non-smz.31

• Pω2 and even all quotientsPω2/Pα preserves Lebesgue outer measure positivity.32

• Pω2 preservesℵ1 and satisfies theℵ2-cc.33

ThenPω2 forces BC.34

Remark 1.5. It is well-known that both Laver reals and random reals preserve positivity (see Lemma 2.28).35

As Laver forcing makes every old uncountable set non-smz, weconclude that a countable support iteration36

of lengthω2 of Laver reals, or alternatively, a countable support iteration alternating Laver with random37

reals, forces BC. The latter iteration also forces the failure of dBC, since the random reals increase the38

covering number of the null ideal, and every set smaller thanthis cardinal is sm.1

4This is a lie, and moreover a stupid (i.e., useless) lie. It isa lie, since we only get something like: for one random over a specificmodel, we can find a systemD such thatLD preserves randomness. It is a useless lie, since preservation of positivity is not enoughanyway: We need a stronger property that is preserved under proper countable support iterations.

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1.C. A preparatory forcing for a single step. Let us first describe how to generically create a single2

forcing, e.g., an ultralaver forcing.3

Let Q be a forcing,M a countable transitive model,P ∈ M a subforcing ofQ. We say thatP is an4

M-complete subforcing ofQ, if every maximal antichainA ∈ M of P is also a maximal antichain inQ. In5

this case everyQ-generic filter overV induces aP-generic filter overM.6

Let Mx be a (countable) model, andDx≔ (Dx

s)s∈ω<ω a system of ultrafilters inMx. This defines the7

ultralaver forcingQx = LDx in Mx. Given any systemD of ultrafilters (inV) such that eachDs extends8

Dxs, then we can show thatQx is anMx-complete subforcing of the ultralaver forcingQ ≔ LD in V. We9

describe this by5 “( Mx,Qx) canonically embeds into Q”.10

So everyQ-generic filterH overV induces aQx-generic filter overMx which we callHx. A trivial but11

crucial observation is the following: When we evaluate the ultralaver real forQ in V[H] then we get the12

same real as when we evaluate it forQx in Mx[Hx]. Of courseQx is not a complete subforcing ofQ, just13

an Mx-complete one: WhileQx is just countable, therefore equivalent to Cohen forcing (from the point14

of view of V), the real added by theQx-genericHx is an ultralaver real (overMx as well as overV), and15

therefore does not add a Cohen real overV.16

We can define a preparatory forcingRL (for a single ultralaver forcing) consisting of pairsx = (Mx,Qx)17

as above (M in someH(χ∗), say), and ordered as follows:y is stronger thanx if Mx ∈ My and (My18

thinks that) (Mx,Qx) canonically embeds intoQy. It is not hard to see thatRL is σ-closed, and adds in the19

extension a generic ultralaver forcingQ such that eachx in the generic filter embeds intoQ.20

Let G beRL-generic (overV). So inV[G], we know thatQx is anMx-complete subforcing ofQ for all21

x ∈ G. Let H beQ-generic (overV[G]). ThenH induces aQx-generic filter overMx (which we callHx)22

for all x ∈ G. Let us repeat the trivial observation: As above, eachMx[Hx] will see the “real” ultralaver23

real (i.e., the one ofV[G][H]).24

Note that “canonical embedding” is a form of approximation:If x is in the generic filter, we do not25

know everything aboutQ, but we know thatQx ⊆ Q and that the maximal antichains ofQx in Mx will be26

maximal antichains inQ as well.27

We should at this stage mention another simple concept that will be used several times: Given (Mx,Qx)28

and (inV) some ultralaver forcingQ such that (Mx,Qx) embeds intoQ, we can take some countable ele-29

mentary submodelN of H(χ∗) containing (Mx,Qx) andQ, and Mostowski-collapse (N,Q) to y = (My,Qy).30

Theny is inRL and stronger thanx.31

1.D. Janus forcing. With “Janus forcings” we denote a family of forcing notions (as in the case of “ul-32

tralaver forcings”). Every Janus forcingJ is a subset ofH(ℵ1) and has a countable “core”∇ (which is the33

same for every Janus forcing) and some additional “stuffing”.6 The forcing∇will add a generic real (“Janus34

real”) coding a null setZ∇. The forcing∇will not be a complete subforcing ofJ, but we will require that all35

maximal antichains involved in the nameZ∇ are also maximal inJ, so thatJ also adds a generic null setZ∇.36

The crucial combinatorial content of Janus forcings heavily relies on previous work by Bartoszynski37

and Shelah [BS10].38

Analogously to the case of ultralaver forcing, letRJ consist of pairsx = (Mx,Qx) such thatMx is a39

countable model andQx a Janus forcing inMx. Given x ∈ RJ and a Janus forcingQ in V, we say thatx40

canonically embeds intoQ if Qx is anMx-complete subforcing ofQ, and we sety ≤ x in RJ if My thinks41

that x canonically embeds intoQy. AgainRJ is aσ-closed forcing and adds a generic objectQ that is42

(forced to be) a Janus forcing; and for everyx in theRJ-generic filter,x canonically embeds intoQ.43

As in the case of ultralaver forcing, the Janus realZ∇ is absolute.44

Other than in the case of ultralaver forcing, every Janus forcing Qx in any modelMx is itself a Janus45

forcing inV; so (taking the collapse of an elementary submodel as above)we trivially get:1

(1.6)For everyx ∈ RJ there is ay ≤ x such that inMy, Qy is a countable Janus forcing (soin particular equivalent to Cohen forcing).

5Note the linguistic asymmetry here: A symmetric and more verbose variant would say “(Mx,Qx) canonically embeds into(V,Q)”.

6Actually, the definition of Janus forcing additionally depends on a real parameter. In our application, we will use ultralaverforcings as even stagesα, and use a Janus forcing defined from the ultralaver real in the stageα+1. The following claims about Janusforcings only hold for this situation; in particular the ground-model sets mentioned have to live in the model before theultralaverforcing.

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ω2

R

P ∈ V[G] (the generic forcing iteration, direct limit ofG)

Py ∈ My

Px ∈ Mx

Mx-complete

My-complete

y < x in G

x ∈ G

F 1. G is the generic filter for the preparatory forcing notionR, which adds thegeneric iterationP. The forcing that gives BC+dBC isR ∗ Pω2. (Of course, in contrastto the impression given by the diagram, the setMx ∩ω2, which is the “domain” ofPx, isnot an interval.)

A crucial property of Janus forcing is that we can make it intorandom forcing as well:2

(1.7)For everyx ∈ RJ there is ay ≤ x such that inMy, Qy is forcing equivalent to randomforcing.

So here we see an important property of the preparatory forcingRJ, which might seem a bit paradoxical3

at first: “Densely”,Q seems to be Cohen as well as random forcing. This two-faced behavior gives Janus4

forcing its name; one could also describe this behavior as “faking” (faking to be Cohen and faking to be5

random).6

Janus forcing is the forcing notion that replaces the Cohen real in the dBC part of the proof. The crucialpoint is:

A countable Janus forcing makes every uncountable ground model set of reals non-sm.

Well, that is actually not much of a point at all: As Carlson has shown, this is achieved by a Cohen real,7

and obviously a countable Janus forcing is equivalent to a Cohen real. And Carlson even showed: When8

adding a Cohen real, this uncountable ground model set remains non-sm even after forcing with another9

forcing notion, provided this forcing notion has precaliber ℵ1.10

So what we actually claim for Janus forcing is the more “explicit” version of our trivial (after Carlson)11

claim above. First, let us introduce an obvious notation:12

(1.8)Let Z be a null set. We say that “Z witnesses thatX is notsm” if there is no realt with(X + t) ∩ Z = ∅, or equivalently, ifX + Z = 2ω.(ClearlyX is not sm iff there is a witnessZ.)

So what we really claim is the following:13

(1.9)The canonical null setZ∇ added by a countable Janus forcing has the property thatX + Z∇ = 2ω for all uncountable ground model setsX, and moreoverX + Z∇ = 2ω ispreserved by every subsequentσ-centered forcing.

Of courseZ∇ is interpreted as a code for a null set, not a concrete subset of the reals (otherwiseX+Z∇ = 2ω14

could not hold when we add new reals).15

So the point here is that we can construct the null setZ∇ (rather: the code) in an absolute way from the16

Janus real (and not just in a non-canonical way via some equivalence to Cohen forcing that we get out of17

countability).18

1.E. The preparatory forcing for the iteration. The preparatory forcingR that we will use will be similar19

toRL or toRJ, but instead of “approximating” a single generic ultralaver or Janus forcing, we approximate20

the alternating iterationP mentioned in Section 1.A. So our preparatory forcingR consists of pairsx =1

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(Mx, Px), whereMx is a countable model7 (and subset of some fixedH(χ∗)) andPx is in Mx an alternating2

iteration.3

Assume thatx ∈ R and thatP is (inV) an alternating iteration. Other than in the case of a singleultralaver4

forcing, we now cannot formally assume thatPx is a subset ofP, but there is a natural construction that tries5

to give anMx-complete embedding ofPx into P. If this construction works, we say that “x is canonically6

embeddable intoP”, and in that case we can treatPx as subset ofP. So if x is canonically embeddable7

into P, andH is aPω2-generic filter overV (which of course inducesPα-generic filtersHα for all α ≤ ω2),8

we can get a canonicalPxα-generic filter overMx for everyα ∈ ω2 ∩ Mx (which we callHx

α).9

We define the order inR as above: Forx, y ∈ R, we definey to be stronger thanx, if Mx ∈ My andMy10

thinks thatx canonically embeds intoPy.11

Note that whileMx thinks thatPx is an iteration of lengthω2, in V (or in My for y ≤ x) the “real domain”12

of Px is just countable (since it is a subset ofMx).13

As promised, one can show thatR is σ-closed and adds a generic alternating iterationP, and thatP is14

ccc. The final limitPω2 is the direct limit of thePα (and thus does not add any new reals in the last stage).15

The intermediate stages satisfy CH, whilePω2 forces 2ℵ0 = ℵ2. As might be expected by now, eachx in16

theR-generic filterG canonically embeds intoP. We will call theR-generic filterG. (The situation is17

illustrated in Figure 1.) So ifH is Pω2-generic overV[G], then we get canonicalPxω2

-generic filtersHx18

for all x ∈ G; and the “real” ultralaver (and Janus) reals calculated inV[G][H] are the same as the ones19

“locally” calculated inMx[Hx].20

Given anyx ∈ R we can construct (inV) an alternating iterationP such thatx embeds intoP and such21

thatP has either of the following two properties:22

• All Janus forcings are countable, at all stagesα not in Mx we “do nothing”, and all limitsPδ are23

“almost finite support overx” (basically the limit is finite support, but we more or less add the24

countably many elements ofPxδ).

25

• All Janus forcings are equivalent to random forcing, and alllimits Pδ are “almost countable support26

overx” (basically we take all conditions in the countable supportlimit that arex-generic).27

The point is that these iterations behave more or less like finite (or countable) support iterations; but we28

can still embedx into them. For example,Mx could think thatPx is a countable support iteration, but we29

may still chooseP to be an almost finite support iteration.30

“Behave more or less in the same way” implies in particular inthe first case that anyPα is σ-centered:31

We iterate only countably many forcings, since we do nothingoutsideMx; the single forcings areσ-32

centered (in the ultralaver case) and even countable in the Janus case, and the (almost) finite support limits33

preserveσ-centeredness.34

In the second case, we get preservation of positivity (with respect to outer Lebesgue measure): ultralaver35

as well as random forcings preserve positivity, and preservation is preserved by (almost) countable support36

(proper) iterations.837

As above, we put the iterationP into a countable elementary submodel; collapse it, and thusget:38

(1.10)For all x ∈ R there is ay ≤ x such that (My thinks that)Py

ω2is σ-centered and all

Janus forcings are countable.39

(1.11)For all x ∈ R there is ay ≤ x such that (My thinks that)Py

ω2preserves Lebesgue outer

measure positivity.

Let us again note that densely often we use finite support, butwe also use countable support densely40

often.41

1.F. Why BC holds. We want to show that BC is forced byR ∗Pω2. Let X be the name of a set of reals of42

sizeℵ1. SincePω2 has lengthω2, we can assume9 thatX is in the ground modelV. We want to show BC,43

so we have to show thatX is not smz. The following is illustrated by Figure 2.1

7Since we are interested in iterations of lengthω2, we cannot use transitive models, that can only see ordinals< ω1. Instead, weuse ord-transitive models.

8Of course, this is not true, rather we need an iterable property such as preservation of random reals over models, etc. We do notget this stronger property universally, we can just preserve a specific random; so claim (1.11) is a lie, too.

9Well, we can’t. But we can do something similar, as will be explained in section 6.B.

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βα + 1 ω2

Xy + F pos.

F appears atα + 1 Assume R Pβ X + F ⊆ Z

(3) pick x s.t.:

(4) picky ≤ x s.t:

Z is Pxβ-name

(5) contradiction!

so: Xy + F pos.(c) Py pres. pos.

(Xy ⊆ X,F,Z abs.)

x

y

R

Px

Py

P(1) (2)

(b) (d)

F 2. The proof of BC.

(1) Fix any ultralaver positionα. (Well, we fixα large enough to justify our assumption thatX ∈ V.)2

We know that the ultralaver real that is added byQα (i.e., appears at stageα + 1) defines in an3

absolute way a (code for a) closed null setF.4

(2) According to Theorem 1.2, it is enough to show thatX+ F is non-null in the extension byR ∗ Pω25

(where the Borel codeF is evaluated in the extension). So assume towards a contradiction that6

X + F is forced to be a subset of a null set (or rather, a Borel code)Z; this already has to happen107

at some stageβ < ω2. In other words: We assume (towards a contradiction)8

R Pβ X + F ⊆ Z.

(3) SincePβ is (forced to be) ccc, we can find a very “absolute” (countable) name forZ; and we can9

find anx ∈ R that already calculatesZ correctly.1110

(4) Now we construct (inV) ay ≤ x in R (with Pyα proper) that satisfies (1.11), and moreover such that11

Xy≔ X∩My ∈ My is uncountable inMy (we get this for free ifMy is the collapse of an elementary12

submodelN with X ∈ N) In particular, (My thinks that)13

(a) Pyα is proper, thus preservesℵ1, thus forces thatXy is uncountable.14

(b) ThereforeQyα forces thatXy + F is positive (according to (1.3)).15

(c) Pyω2

preserves positivity.16

(d) ThereforePyβ

forces thatXy + F is positive12 (and in particular not a subset ofZ).17

(5) This leads to the obvious contradiction: LetG beR-generic overV and containy, and letHβ be18

Pβ-generic overV[G]. Then Hyβ

is Pyβ-generic overMy, and thereforeMy[Hy

β] thinks that some19

x+ f ∈ Xy + F is not inZ. But Xy = X ∩ My ⊆ X, and the codes forF and forZ are absolute (for20

F since it is constructed in a canonical way from the ultralaver real, forZ because we took care of21

it in step (3)).22

1.G. Why dBC holds. The proof of dBC is similar, usingσ-centeredness instead of positivity preserving,23

and a countable Janus forcing instead of ultralaver forcing.24

Let X be the name of a set of reals of sizeℵ1. Again, without loss of generality13 X ∈ V. We want to25

show dBC, so we have to show thatX is not sm.26

(1) Fix any Janus positionα (large enough to justify our assumption thatX ∈ V). We know that the27

Janus real that is added byQα (i.e., appears at stageα + 1) defines in an absolute way a (code for28

a) null setZ∇.1

10Each real (and in particular the Borel codeZ) in thePω2-extension already has to appear at some stageβ < ω2; and the statement“X + F ⊆ Z” is absolute.

11More formally: We find anx ∈ R and aPxβ-nameZx in Mx such thatx forces (inR) that Pβ forces thatZ (evaluated by the

Pβ-generic) is the same asZx (evaluated by the inducedPxβ-generic).

12Here we even get positivity ofXy + F whereF is evaluated in the intermediate extension of stageα + 1. However, we get thecontradiction even if we just assume thatXy + F is positive whereF is evaluated in thePy

β-extension.

13And again, this is a lie.

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(2) According to (1.8), it is enough to show thatZ∇ + X = 2ω in the extension byR ∗ Pω2. So assume2

towards a contradiction thatZ∇ + X , 2ω. This already happens at some stageβ < ω, i.e., we3

assume4

R Pβ r < Z∇ + X.

(3) Again, findx such thatr is an “absolute”Pxβ-name.

5(4) Now we construct (inV) a y ≤ x in R that satisfies (1.10), and such thatXy

≔ X ∩ My ∈ My is6

uncountable inMy. In particular, (My thinks that)7

(a) Pyα is proper, thus preservesℵ1, and so forces thatXy is uncountable.8

(b) Qyα is (forced to be) a countable Janus forcing notion.9

(c) Pyω2

isσ-centered.10

(d) Therefore (1.9) implies thatPyβ

forces thatZ∇ + Xy = 2ω, in particular thatr ∈ Z∇ + Xy.11

(5) As before, this leads to a contradiction.12

2. U 13

In this section, we define the family ofultralaver forcingsLD, variants of Laver forcing which depend14

on a systemD of ultrafilters. We will need the following properties of this forcing notion:15

Wish List 2.1. (1) LD isσ-centered, hence ccc. (This will follow trivially from the definition.)16

(2) Ultralaver kills smz: There is a canonicalLD-name¯˜ℓ for a fast growing real inωω called the17

ultralaver real. From this real, we can define a closed null set F such thatX + F is positive for all18

uncountableX in V (and thereforeF witnesses thatX is not smz, according to Theorem 1.2). (See19

Corollary 2.20.)20

(3) “Canonical” above in particular means: IfM is a countable model,LDM ∈ M is an ultralaver21

forcing (with ultralaver real˜ℓM) which M-completely embeds into the ultralaver forcingLD (with22

ultralaver real˜ℓ), then in theLD-extensionV[H], ¯

˜ℓ[H] is the same as

˜ℓM[HM], whereHM is the23

induced filter onLDM , similarly for F.24

(4) WheneverX is uncountable, thenLD forces thatX is not “thin”. (See Corollary 2.23.)25

(5) If (M, ∈) is a countable model of ZFC* and ifLDM is an ultralaver forcing inM (with ultralaver26

real ¯˜ℓM), then for any ultrafilter systemD extendingDM, LDM is anM-complete subforcing of the27

ultralaver forcingLD (which adds the same ultralaver real). (See Lemma 2.4.)28

(6) Moreover, givenM andLDM as above, and a random realr over M, we can chooseD extending29

DM such thatLD forces that randomness ofr is preserved (in a strong way that can be preserved in30

a countable support iteration). (See Lemma 2.29.)31

(Advice to the reader: At first reading, you may want to take these properties as granted, and skip to32

Section 3.)33

2.A. Definition of ultralaver.34

Notation. We use the following fairly standard notation:35

A treeis a nonempty setp ⊆ ω<ω which is closed under initial segments and has no maximal elements.1436

The elements (“nodes”) of a tree are partially ordered by⊆.37

For each sequences ∈ ω<ω we write lh(s) for the length ofs.38

For any treep ⊆ ω<ω and anys ∈ p we write succp(s) for one of the following two sets:39

k ∈ ω : sk ∈ p or t ∈ p : (∃k ∈ ω) t = sk

and we rely on the context to help the reader decide which set we mean.40

A branchof p is either of the following:41

• A function f : ω→ ω with fn ∈ p for all n ∈ ω.42

• A maximal chain in the partial order (p,⊆). (As our trees do not have maximal elements, each43

such chainC determines a branch⋃

C in the first sense, and conversely.)1

14Except for the proof of Lemma 2.4, where we also allow trees with maximal elements, and even empty trees.

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We write [p] for the set of all branches ofp.2

For any treep ⊆ ω<ω and anys ∈ p we write p[s] for the sett ∈ p : t ⊇ sor t ⊆ s, and we write [s] for3

either of the following sets:4

t ∈ p : s⊆ t or x ∈ [p] : s⊆ x.

The stem of a treep is the shortests ∈ p with | succp(s)| > 1. (The trees we consider will never be5

branches, i.e., will always have finite stems.)6

Definition 2.2. • For treesq, p we writeq ≤ p if q ⊆ p (“q is stronger thanp”), and we say that “q7

is a pure extension of p” (q ≤0 p) if q ≤ p and stem(q) = stem(p).8

• A filter systemD is a family (Ds)s∈ω<ω of filters onω. (All our filters will contain the Frechet filter9

of cofinite sets.) We writeD+s for the collection ofDs-positive sets (i.e., sets whose complement is10

not in Ds).11

• We defineLD as the set of all treesp such that succp(t) ∈ D+t for all t ∈ p above the stem.12

• The generic filter is determined by the generic branchℓ = (ℓi)i∈ω ∈ ωω, called thegeneric real:13

ℓ =⋂

p∈G[p] or equivalently,ℓ =⋃

p∈G stem(p).14

• An ultrafilter system is a filter system consisting of ultrafilters. (Since all our filters contain the15

Frechet filter, we only consider nonprincipal ultrafilters.)16

• An ultralaver forcing is a forcingLD defined from an ultrafilter system. The generic real for an17

ultralaver forcing is also called theultralaver real.18

Recall that a forcing notion (P,≤) is σ-centerediff P =⋃

n Pn, where for alln, k ∈ ω and for all19

p1, . . . , pk ∈ Pn there isq ≤ p1, . . . , pk.20

Note that all ultralaver forcingsLD areσ-centered (fulfilling our first wish 2.1(1)), as finite set of con-21

ditions with the same stem has a common lower bound. If eachDs is the Frechet filter, thenLD is Laver22

forcing (often just writtenL).23

2.B. M-complete embeddings.Note that for all ultrafilter systemsD we have:24

(2.3)Two conditions inLD are compatible if and only if their stems are comparable andmoreover, the longer stem is an element of the condition withthe shorter stem.

Lemma 2.4. Let M be countable.15 In M, letLDM be an ultralaver forcing. LetD be (in V) a filter system25

extending16 DM. ThenLDM is an M-complete subforcing ofLD.26

Proof. For any tree17 T, any filter systemE = (Es)s∈ω<ω , and anys0 ∈ T we define a sequence (TαE,s0

)α∈ω127

of “derivatives” (where we may abbreviateTαE,s0

to Tα) as follows:28

• T0≔ T [s0].29

• GivenTα, we letTα+1≔ Tα \

⋃

[s] : s ∈ Tα, s0 ⊆ s, succTα (s) < E+s , where [s] ≔ t : s⊆ t.30

• For limit ordinalsδ > 0 we letTδ ≔⋂

α<δ Tα.31

Then we have32

(a) EachTα is closed under initial segments. Also:α < β impliesTα ⊇ Tβ.33

(b) There is anα0 < ω1 such thatTα0 = Tα0+1 = Tβ for all β > α0. We writeT∞ or T∞E,s0

for Tα0.34

(c) If s0 ∈ T∞E,s0

, thenT∞E,s0∈ LE with stems0.35

Conversely, if stem(T) = s0, andT ∈ LE, thenT∞ = T.36

(d) If T contains a treeq ∈ LE with stem(q) = s0, thenT∞ containsq∞ = q, so in particulars0 ∈ T∞.37

(e) Thus:T contains a condition inLE with stems0 iff s0 ∈ T∞E,s0

.38

(f) The computation ofT∞ is absolute between any two models containingT andE. (In particular,39

any transitive ZFC*-model containingT will also containα0.)1

15Here, we can assume thatM is a countable transitive model of a sufficiently large finite subset ZFC* of ZFC. Later, we will alsouse ord-transitive models instead of transitive ones, which does not make any difference as far as properties ofLD are concerned, asour arguments take place in transitive parts of such models.

16I.e.,DMs ⊆ Ds for all s ∈ ω<ω.

17Here we also allow empty trees, and trees with maximal nodes.

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(g) Moreover: LetT ∈ M, E ∈ M, and letE′ be a filter system extendingE such that for alls0 and2

all A ∈ P(ω) ∩ M we have:A ∈ (Es0)+ iff A ∈ (E′s0

)+. (In particular, this will be true for anyE′3

extendingE, provided that eachEs0 is anM-ultrafilter.)4

Then for eachα ∈ M we haveTαE,s0= Tα

E′ ,s0(and henceTα

E,s0∈ M). (Proved by induction onα.)

5

Now let A = (pi : i ∈ I ) ∈ M be a maximal antichain inLDM , and assume (inV) that q ∈ LD. Let6

s0 ≔ stem(q).7

We will show thatq is compatible with somepi . This is clear if there is somei with s0 ∈ pi and8

stem(pi) ⊆ s0, by (2.3). (In this case,pi ∩ q is a condition inLD with stems0.)9

So for the rest of the proof we assume that this is not the case,i.e.:10

(2.5) There is noi with s0 ∈ pi and stem(pi) ⊆ s0.

Let J ≔ i ∈ I : s0 ⊆ stem(pi). We claim that there isj ∈ J with stem(p j) ∈ q (which as above implies11

thatq andp j are compatible).12

Assume towards a contradiction that this is not the case. Then q is contained in the following treeT:

T ≔ (ω<ω)[s0] \⋃

j∈J

[stem(p j)](2.6)

Note thatT ∈ M. In V we have:13

(2.7) The treeT contains a conditionq with stems0.

so by (e) (applied inV), followed by (g), and again by (e) (now inM) we get:14

(2.8) The treeT also contains a conditionp ∈ M with stems0.

Now p has to be compatible with somepi . The sequencess0 = stem(p) and stem(pi) have to be comparable,15

so by (2.3) there are two possibilities:16

(1) stem(pi) ⊆ stem(p) = s0 ∈ pi . We have excluded this case in our assumption (2.5).17

(2) s0 = stem(p) ⊆ stem(pi) ∈ p. Soi ∈ J. By construction ofT (see (2.6)), we conclude stem(pi) < T,18

contradicting stem(pi) ∈ p ⊆ T (see 2.8). 19

2.C. Ultralaver kills strong measure zero. The proof of the following lemma is a well-known routine20

construction that works with many tree forcings. We will delay the proof until Lemma 2.34.21

Lemma 2.9. If A is a finite set,˜α anLD-name, p∈ LD, and p

˜α ∈ A, then there isβ ∈ A and a pure22

extension q≤0 p such that q ˜α = β.23

Definition 2.10. Let ℓ be an increasing sequence of natural numbers. We say thatX ⊆ 2ω is smz with24

respect toℓ, if there exists a sequence (Ik)k∈ω of basic intervals of 2ω of measure≤ 2−ℓk (i.e., eachIk is of25

the form [sk] for somesk ∈ 2ℓk) such thatX ⊆⋂

m∈ω⋃

k≥m Ik.26

Remark 2.11. It is well known and easy to see that the properties27

• For all ℓ there exists exists a sequence (Ik)k∈ω of basic intervals of 2ω of measure≤ 2−ℓk such that28

X ⊆⋃

k∈ω Ik.29

• For all ℓ there exists exists a sequence (Ik)k∈ω of basic intervals of 2ω of measure≤ 2−ℓk such that30

X ⊆⋂

m∈ω⋃

k≥m Ik.31

are equivalent. Hence, a setX is smz iff X is smz with respect to allℓ ∈ ωω.32

The following lemma is a variant of the corresponding lemma (and proof) for Laver forcing (see for33

example [Jec03, Lemma 28.20]): Ultralaver makes old uncountable sets non-smz.34

Lemma 2.12. Let D be a system of ultrafilters, and let¯˜ℓ be theLD-name for the ultralaver real. Then each35

uncountable set X∈ V is forced to be non-smz (witnessed by the ultralaver real¯˜ℓ).36

More precisely, the following holds:37

(2.13) LD∀X ∈ V ∩ [2ω]ℵ1 ∀(xk)k∈ω ⊆ 2ω X *

⋂

m∈ω

⋃

k≥m

[xk˜ℓk].

We first give two technical lemmas:1

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Lemma 2.14. Let p∈ LD with stem s∈ ω<ω, and let˜x be aLD-name for a real in2ω. Then there exists a2

pure extension q≤0 p and a realτ ∈ 2ω such that for every n∈ ω,3

(2.15) i ∈ succq(s) : q[si] ˜xn = τn ∈ Ds.

Proof. For eachi ∈ succp(s), let qi ≤0 p[s i] be such thatqi decides˜xi, i.e., there is ati of lengthi such4

thatqi ˜xi = ti (this is possible by Lemma 2.9).5

Now we define the realτ ∈ 2ω as theDs-limit of the ti ’s. In more detail: For eachn ∈ ω there is a6

(unique)τn ∈ 2n such thati : tin = τn ∈ Ds; sinceDs is a filter, there is a realτ ∈ 2ω with τn = τn for7

eachn. Finally, letq≔⋃

i qi . 8

Lemma 2.16. Let p ∈ LD with stem s, and let(˜xk)k∈ω be a sequence ofLD-names for reals in2ω. Then9

there exists a pure extension q≤0 p and a family of reals(τη)η∈q, η⊇s ⊆ 2ω such that for eachη ∈ q above s,10

and every n∈ ω,11

(2.17) i ∈ succq(η) : q[ηi] ˜x|η|n = τηn ∈ Dη.

Proof. We apply Lemma 2.14 to each nodeη in p aboves (and to˜x|η|) separately: We first get ap1 ≤0 p12

and aτs ∈ 2ω; for every immediate successorη ∈ succp1(s), we getqη ≤0 p[η]1 and aτη ∈ 2ω, and let13

p2 ≔⋃

η qη; in this way, we get a (fusion) sequence (p, p1, p2, . . .), and letq≔⋂

k pk. 14

Proof of Lemma 2.12.We want to prove (2.13). Assume towards a contradiction thatX is an uncountable15

set inV, and that (˜xk)k∈ω is a sequence of names for reals in 2ω andp ∈ LD such that16

(2.18) p X ⊆⋂

m∈ω

⋃

k≥m

[˜xk

˜ℓk].

Let s ∈ ω<ω be the stem ofp.17

By Lemma 2.16, we can fix a pure extensionq ≤0 p and a family (τη)η∈q, η⊇s ⊆ 2ω such that for each18

η ∈ q above the stems and everyn ∈ ω, condition (2.17) holds.19

SinceX is (in V and) uncountable, we can find a realx∗ ∈ X which is different from each real in the20

countable family (τη)η∈q, η⊇s; more specifically, we can pick a family of natural numbers (nη)η∈q, η⊇s such21

thatx∗nη , τηnη for anyη.22

We can now findr ≤0 q such that:23

• For allη ∈ r abovesand alli ∈ succr (η) we havei > nη.24

• For allη ∈ r abovesand alli ∈ succr (η) we haver [η i] ˜x|η|nη = τηnη , x∗nη.25

So for allη ∈ r aboveswe have, writingk for |η|, thatr [η i] forcesx∗ < [˜xknη] ⊇ [

˜xkℓk]. We conclude26

thatr forcesx∗ <⋃

k≥|s|[˜xkℓk], contradicting (2.18). 27

Corollary 2.19. Let (tk)k∈ω be a dense subset of2ω.28

Let D be a system of ultrafilters, and let˜ℓ be theLD-name for the ultralaver real. Then the set

˜H ≔

⋂

m

⋃

k≥m

[tk˜ℓk]

is forced to be a comeager set with the property that˜H does not contain any translate of any old uncount-29

able set.30

Using Pawlikowski’s Theorem 1.2 we get:31

Corollary 2.20. There is a canonical name F for a closed null set such that X+ F is positive for all32

uncountable X in V.33

In particular, no uncountable ground model set is smz in the ultralaver extension.34

2.D. Thin sets and strong measure zero.For the notion of “(very) thin” set, we use an increasing func-35

tion B∗(k) (the function we use will be described in Corollary 3.3). Wewill assume thatℓ∗ = (ℓ∗k)k∈ω is36

an increasing sequence of natural numbers withℓ∗k+1 ≫ B∗(k). (We will later use a subsequence of the37

ultralaver realℓ asℓ∗, see Lemma 2.22).38

Definition 2.21. For X ⊆ 2ω andk ∈ ω we writeX[ℓ∗k, ℓ∗k+1) for the setx[ℓ∗k, ℓ

∗k+1) : x ∈ X. We say that

39

• X ⊆ 2ω is “very thin with respect toℓ∗ and B∗”, if there are infinitely manyk with |X[ℓ∗k, ℓ∗k+1)| ≤40

B∗(k).1

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• X ⊆ 2ω is “thin with respect toℓ∗ and B∗”, if X is the union of countably many very thin sets.2

Note that the family of thin sets is aσ-ideal, while the family of very thin sets is not even an ideal. Also,3

every very thin set is covered by a closed very thin (in particular nowhere dense) set. In particular, every4

thin set is meager and the ideal of thin sets is a proper ideal.5

Lemma 2.22. Let B∗ be an increasing function. Letℓ be an increasing sequence of natural numbers. We6

define a subsequenceℓ∗ of ℓ in the following way:ℓ∗k = ℓnk where nk+1 − nk = B∗(k) · 2ℓ∗k .7

Then we get: If X is thin with respect toℓ∗ and B∗, then X is smz with respect toℓ.8

Proof. Assume thatX =⋃

i∈ω Yi , eachYi very thin with respect toℓ∗ andB∗. Let (X j) j∈ω be an enumeration9

of Yi : i ∈ ω where eachYi appears infinitely often. SoX ⊆⋂

m∈ω⋃

j≥m X j .10

By induction onj ∈ ω, we find for all j > 0 somek j > k j−1 such that11

|X j[ℓ∗k j, ℓ∗k j+1)| ≤ B∗(k j) hence |X j[0, ℓ∗k j+1)| ≤ B∗(k j) · 2

ℓ∗kj = nk j+1 − nk j .

So we can enumerateX j[0, ℓ∗k j+1) as (si)nkj≤i<nkj+1. HenceX j is a subset of⋃

nkj≤i<nkj+1[si ]; and eachsi has12

lengthℓ∗k j+1 ≥ ℓi , sinceℓ∗k j+1 = ℓnkj+1 andi < nk j+1. This implies13

X ⊆⋂

m∈ω

⋃

j≥m

X j ⊆⋂

m∈ω

⋃

i≥m

[si ]

HenceX is smz with respect toℓ. 14

Lemma 2.12 and Lemma 2.22 yield:15

Corollary 2.23. Let B∗ be an increasing function. LetD be a system of ultrafilters, and˜ℓ the name for the16

ultralaver real. Let˜ℓ∗ be constructed from B∗ and

˜ℓ as in Lemma 2.22.17

ThenLD forces that for every uncountable X⊆ 2ω:18

• X is not smz with respect to˜ℓ.19

• X is not thin with respect to˜ℓ∗ and B∗.20

2.E. Ultralaver and preservation of Lebesgue positivity. It is well known that both Laver forcing and21

random forcing preserve Lebesgue positivity; in fact they satisfy a stronger property that is preserved22

under countable support iterations. (So in particular, a countable support iteration of Laver and random23

also preserves positivity.)24

Ultralaver forcingLD will in general not preserve positivity. Indeed, if all ultrafiltersDs are equal to the25

same ultrafilterD∗, then the rangeL ≔ ℓ0, ℓ1, . . . ⊆ ω of the ultralaver realℓ will diagonalizeD∗, so every26

ground model realx ∈ 2ω (viewed as a subset ofω) will either almost containL or be almost disjoint toL,27

which implies that the set 2ω ∩ V of old reals is covered by a null set in the extension. However, later in28

this paper it will become clear that if we choose the ultrafiltersDs in a sufficiently generic way, then many29

old positive sets will stay positive. More specifically, in this section we will show (Lemma 2.29): IfDM is30

we will show how a in a countable modelM andr a random real overM, then we can find an extensionD31

such thatLD forces thatr remains random overM[HM] (and additionally some “side conditions” are met,32

which are necessary to preserve the property in forcing iterations).33

In Section 4.D we will see how to use this property to preserver randoms in limits.34

The setup we use for preservation of randomness is basicallythe notation of “Case A” preservation35

introduced in [She98, Ch.XVIII], see also [Gol93, GK06] or the textbook [BJ95, 6.1.B]:36

Definition 2.24. We writeΩ for the collection of clopen sets on 2ω. We say that the functionZ : ω → Ω37

is a code for a null set, if the measure ofZ(n) is at most 2−n for eachn ∈ ω.38

For such a codeZ, the set nullset(Z) coded byZ is39

nullset(Z) ≔⋂

n

⋃

k≥n

Z(k).

The set nullset(Z) obviously is a null set, and it is well known that every null set is contained in such a40

set nullset(Z).1

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Definition 2.25. For a realr and any codeZ, we defineZ ⊏n r by:2

(∀k ≥ n) r < Z(k).

We writeZ ⊏ r if Z ⊏n r holds for somen; i.e., if r < nullset(Z).3

For later reference, we record the following trivial fact:4

(2.26) p ˜Z ⊏ r iff there is a name

ñ for an element ofω such thatp

˜Z ⊏

ñ r.

Let P be a forcing notion, and˜Z a P-name of a code for a null set. An interpretation of

˜Z below p is5

some codeZ∗ such that there is a sequencep = p0 ≥ p1 ≥ p2 ≥ . . . such thatpm forces˜Zm = Z∗m.6

Usually we demand (which allows a simpler proof of the preservation theorem at limit stages) that the7

sequence (p0, p1, . . . ) is inconsistent, i.e.,p forces that there is anmsuch thatpm < G. Note that whenever8

P adds a newω-sequence of ordinals, we can find such an interpretation forany˜Z.9

We now turn to preservation of Lebesgue positivity:10

Definition 2.27. (1) A forcing notionP preserves Borel outer measure, if P forces Leb∗(AV) = Leb(AV[GP])11

for every code for a Borel setA. (Leb∗ denotes the outer Lebesgue measure, and for a Borel code12

A and a set-theoretic universeV, AV denotes the Borel set coded byA in V.)13

(2) P strongly preserves randoms, if the following holds: LetN ≺ H(χ∗) be countable for a sufficiently14

large regular cardinalχ∗, let P, p,˜Z ∈ N, let p ∈ P and letr be random overN. Assume that inN,15

Z∗ is an interpretation of˜Z, and assumeZ∗ ⊏0 r. Then there is anN-genericq ≤ p forcing thatr is16

still random overN[G] and moreover,˜Z ⊏0 r. (In particular,P has to be proper.)17

(3) Assume thatP is absolutely definable.P strongly preserves randoms over countable modelsif (2)18

holds for all countable (transitive18) modelsN of ZFC*.19

It is easy to see that these properties are increasing in strength. (Of course (3)⇒(2) works only if ZFC*20

is satisfied inH(χ∗).)21

In [KS05] it is shown that (1) implies (3), provided thatP is nep (“non-elementary proper”, i.e., nicely22

definable and proper with respect to countable models). In particular, every Suslin ccc forcing notion such23

as random forcing, and also many tree forcing notions including Laver forcing, are nep. HoweverLD is not24

nicely definable in this sense, as its definition uses ultrafilters as parameters.25

Lemma 2.28. Both Laver forcing and random forcing strongly preserve randoms over countable models.26

Proof. For random forcing, this is easy and well known (see, e.g., [BJ95, 6.3.12]).27

For Laver forcing: By the above, it is enough to show (1). Thiswas done by Woodin (unpublished) and28

Judah-Shelah [JS90]. A nicer proof (including a variant of (2)) is given by Pawlikowski [Paw96b]. 29

Ultralaver will generally not preserve Lebesgue positivity, let alone randomness. However, we get30

the following “local” variant of strong preservation of randoms (which will be used in the preservation31

theorem 4.33). The rest of this section will be devoted to theproof of the following lemma.32

Lemma 2.29. Assume that M is a countable model,DM an ultrafilter system in M and r a random real33

over M. Then there is (in V) an ultrafilter systemD extending19 DM, such that the following holds:34

If35

• p ∈ LDM ,36

• in M,˜Z = (

˜Z1, . . . ,

˜Zm) is a sequence ofLDM -names for codes for null sets,20 and Z∗1, . . . ,Z

∗m are37

interpretations under p, witnessed by a sequence(pn)n∈ω with strictly increasing stems,38

• r is random over M,39

• Z∗i ⊏ki r for i = 1, . . . ,m,40

then there is a q≤ p in LD forcing that41

• r is random over M[GM],42

•˜Zi ⊏ki r for i = 1, . . . ,m.1

18later we will introduce ord-transitive models, and it is easy to see that it does not make any difference whether we demandtransitive or not; this can be seen using a transitive collapse

19This implies, by Lemma 2.4, that theLD-generic filterG induces anLDM -generic filter overM, which we callGM .20Recall that nullset(

˜Z) =

⋂

n⋃

k≥n ˜Z(k) is a null set in the extension.

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For the proof of this lemma, we will use the following concepts:2

Definition 2.30. Let p ⊆ ω<ω be a tree. A “front name below p” is a function21 h : F → Ω, whereF ⊆ p3

is a front (a set that meets every branch ofp in a unique point). (For notational simplicity we also allowh4

to be defined on elements< p; this way, every front name belowp is also a front name belowq whenever5

q ≤ p.)6

If h is a front name andD is any filter system, we define the correspondingLD-name (in the sense offorcing)

˜zh by

˜zh≔ (y, p[s]) : s ∈ F, y ∈ h(s).(2.31)

(This does not depend on theD we use, since we set ˇy≔ (x, ω<ω) : x ∈ y.)7

Up to forced equality, the name˜zh is characterized by the fact thatp[s] forces (in anyLD) that

˜zh = h(s),8

for everys in the domain ofh.9

Note that the same objecth can be viewed as a front name belowp with respect to different forcings10

LD1, LD2

, as long asp ∈ LD1∩ LD2

.11

Definition 2.32. Let p ⊆ ω<ω be a tree. A “continuous name below p” is either of the following:12

• An ω-sequence of front names belowp.13

• A ⊆-increasing functiong : p→ Ω<ω such that limn→∞ lh(g(cn)) = ∞ for every branchc ∈ [p].14

For eachn, the set of minimal elements ins ∈ p : lh(g(s)) > n is a front, so each continuous name in the15

second sense naturally defines a name in the first sense, and conversely. Being a continuous name belowp16

does not involve the notion of nor does it depend on the filter systemD.17

If g is a continuous name andD is any filter system, we can again define the correspondingLD-name

˜Zg (in the sense of forcing); we leave a formal definition of

˜Zg to the reader and content ourselves with this

characterization:

(∀s ∈ p) : p[s] LDg(s) ⊆

˜Zg.(2.33)

Note that a continuous name belowp naturally corresponds to a continuous functionF : [p] → Ωω, and18

˜Zg is forced (byp) to be the value ofF at the generic real

˜ℓ.19

Lemma 2.34. LD has the following “pure decision properties”:20

(1) Whenever˜y is a name for an element ofΩ, p ∈ LD, then there is a pure extension p1 ≤0 p such21

that˜y =

˜zh (is forced) for a front name h below p1.

22(2) Whenever

˜Y is a name for a sequence of elements ofΩ, p ∈ LD, then there is a pure extension23

q ≤0 p such that˜Y =

˜Zg (is forced) for some continuous name g below q.24

(3) (This is Lemma 2.9.) If A is a finite set,˜α a name, p∈ LD, and p forces

˜α ∈ A, then there isβ ∈ A25

and a pure extension q≤0 p such that q ˜α = β.26

Proof. Let p ∈ LD, s0 ≔ stem(p),˜y a name for an element ofΩ.

27

We call t ∈ p a “good node inp” if˜y is a front name belowp[t] (more formally: forced to be equal to

˜zh

28

for a front nameh). We can findp1 ≤0 p such that for allt ∈ p1 aboves0: If there isq ≤0 p[t]1 such thatt is29

good inq, thent is already good inp1.30

We claim thats0 is now good (inp1). Note that for any bad nodes the set t ∈ succp1(s) : t bad is31

in D+s . Hence, ifs0 is bad, we can inductively constructp2 ≤0 p1 such that all nodes ofp2 are bad nodes32

in p1. Now letq ≤ p2 decide˜y, s≔ stem(q). Thenq ≤0 p[s]

1 , sos is good inp1, contradiction. This finishes33

the proof of (1).34

To prove (2), we first constructp1 as in (1) with respect to˜y0. This gives a frontF1 ⊆ p1 deciding

˜y0.35

Above each node inF1 we now repeat the construction from (1) with respect to˜y1, yieldingp2, etc. Finally,36

q≔⋂

n pn.37

To prove (3): Similar to (1), we can findp1 ≤0 p such that for eacht ∈ p1: If there is a pure extension38

of p[t]1 deciding

˜α, thenp[t]

1 decides˜α; in this case we again callt good. Since there are only finitely many39

possibilities for the value of˜α, any bad nodet hasD+t many bad successors. So if the stem ofp1 is bad, we40

can again reach a contradiction as in (1). 1

21Instead ofΩ, the set of clopen sets, we may also consider other ranges of front names, such as the class of all ordinals, or thesetω.

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Corollary 2.35. Let D be a filter system, and let G⊆ LD be generic. Then every Y∈ Ωω in V[G] is the2

evaluation of a continuous name˜Zg by G.3

Proof. Assume thatp ∈ LD forces that˜Y is not the evaluation of a continuous name. We can findq ≤0 p4

and a continuous nameg belowq such thatq ˜Y =

˜Zg. This is a contradiction. 5

We will need the following modification of the concept of “continuous names”.6

Definition 2.36. Let p ⊆ ω<ω be a tree,b ∈ [p] a branch. An “almost continuous name below p (with7

respect to b)” is a ⊆-increasing functiong : p → Ω<ω such that limn→∞ lh(g(cn)) = ∞ for every branch8

c ∈ [p], except possibly forc = b.9

Note that “except possibly forc = b” is the only difference between this definition and the definition of10

a continuous name.11

Since for anyD it is forced22 that the generic real (forLD) is not equal to the exceptional branchb, we12

again get a name˜Zg of a function inΩω satisfying:13

(∀s ∈ p) : p[s] LDg(s) ⊆

˜Zg.

An almost continuous name naturally corresponds to a continuous functionF from [p] \ b intoΩω.14

Note that being an almost continuous name is a very simple combinatorial property ofg which does15

not depend onD, nor does it involve the notion . Thus, the same functiong can be viewed as an almost16

continuous name for two different forcing notionsLD1, LD2

simultaneously.17

Lemma 2.37. Let D be a system of filters (not necessarily ultrafilters).18

Assume thatp = (pn)n∈ω witnesses that Y∗ is an interpretation of˜Y, and that the lengths of the stems of19

the pn are strictly increasing.23 Then there exists a sequenceq = (qn)n∈ω such that20

(1) q0 ≥ q1 ≥ · · · .21

(2) qn ≤ pn for all n.22

(3) q also interprets˜Y as Y∗. (This follows from the previous two statements.)23

(4)˜Y is almost continuous below q0, i.e., there is an almost continuous name g such that q0 forces24

˜Y =

˜Zg.)25

(5)˜Y is almost continuous below qn, for all n. (This follows from the previous statement.)26

Proof. Let b be the branch described by the stems of the conditionspn:27

b≔ s : (∃n) s⊆ stem(pn).

We now construct a conditionq0. For everys ∈ b satisfying stem(pn) ⊆ s ( stem(pn+1) we set28

succq0(s) = succpn(s), and for allt ∈ succq0(s) except for the one inb we letq[t]0 ≤0 p[t]

n be such that˜Y is29

continuous belowq[t]0 . We can do this by Lemma 2.34(2).

30

Now we set31

qn ≔ pn ∩ q0 = q[stem(pn)]0 ≤ pn.

This takes care of (1) and (2). Now we show (4): Any branchc of q0 not equal tob must contain a node32

sk < b with s ∈ b, soc is a branch inq[sk]0 , below which

˜Y was continuous.

33

The following lemmas and corollaries are the motivation forconsidering continuous and almost contin-34

uous names.35

Lemma 2.38. Let D be a system of filters (not necessarily ultrafilters). Let p∈ LD, let b be a branch,36

and let g: p→ Ω<ω be an almost continuous name below p with respect to b; write˜Zg for the associated37

LD-name.38

Let r ∈ 2ω be a real, n0 ∈ ω. Then the following are equivalent:39

(1) p LDr <⋃

n≥n0 ˜Zg(n), i.e.,

˜Zg ⊏n0 r.40

(2) For all n ≥ n0 and for all s∈ p for which g(s) has length> n we have r< g(s)(n).41

Note that (2) does not mention the notion and does not depend onD.1

22 This follows from our assumption that all our filters containthe Frechet filter.23It is easy to see that for everyLD-name

˜Y we can find such ¯p andY∗: First find p which interprets both

˜Y and ¯

˜ℓ, and then thin

out to get a strictly increasing sequence of stems.

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Proof. ¬(2)⇒ ¬(1): Assume that there iss ∈ p for which g(s) = (C0, . . . ,Cn, . . . ,Ck) andr ∈ Cn. Then2

p[s] forces that the generic sequence˜Zg = (

˜Z(0),

˜Z(1), . . .) starts withC0, . . . ,Cn, sop[s] forcesr ∈

˜Zg(n).3

¬(1)⇒ ¬(2): Assume thatp does not forcer <⋃

n≥n0 ˜Zg(n). So there is a conditionq ≤ p and some4

n ≥ n0 such thatq r ∈˜Zg(n). By increasing the stem ofq, if necessary, we may assume thats≔ stem(q)5

is not onb (the “exceptional” branch), and thatg(s) has already length> n. Let Cn ≔ g(s)(n) be then-th6

entry ofg(s). So p[s] already forces˜Zg(n) = Cn; now q[s] ≤ p[s], andq[s] forces the following statements:7

r ∈˜Zg(n),

˜Zg(n) = Cn. Hencer ∈ Cn, so (2) fails. 8

Corollary 2.39. Let D1 andD2 be systems of filters, and assume that p is inLD1∩ LD2

. Let g : p→ Ω<ω9

be an almost continuous name of a sequence of clopen sets, andlet˜Zg

1 and˜Zg

2 be the associatedLD1-name10

andLD2-name, respectively.11

Then for any real r and n∈ ω we have12

p LD1 ˜Zg

1 ⊏n r ⇔ p LD2 ˜Zg

2 ⊏n r.

(We will use this corollary for the special case thatLD1is an ultralaver forcing, andLD2

is Laver forcing.)13

Lemma 2.40. Let D1 andD2 be systems of filters, and assume that p is inLD1∩ LD2

. Let g : p→ Ω<ω be14

a continuous name of a sequence of clopen sets, let F⊆ p be a front and let h: F → ω be a front name.15

Again we will write˜Zg

1, ˜Zg

2 for the associated names of codes for null sets, and we will writeñ1 and

ñ2 for16

the associatedLD1- andLD2

-names, respectively, of natural numbers.17

Then for any real r we have:18

p LD1 ˜Zg

1 ⊏ñ1 r ⇔ p LD2 ˜

Zg2 ⊏˜

n2 r.

Proof. Assumep LD1 ˜Zg

1 ⊏ñ1 r. So for eachs ∈ F we have:p[s] LD1 ˜

Zg1 ⊏h(s) r. By Corollary 2.39, we19

also havep[s] LD2 ˜Zg

2 ⊏h(s) r. So alsop[s] LD2 ˜Zg

2 ⊏ñ2 r for eachs ∈ F. Hencep LD2 ˜

Zg2 ⊏˜

n2 r. 20

Corollary 2.41. Assume q∈ L forces in Laver forcing that˜Zgk ⊏ r for k = 1, 2, . . ., where each gk is a21

continuous name of a code for a null set. Then there is a Laver condition q′ ≤0 q such that for all ultrafilter22

systemsD we have:23

If q′ ∈ LD, then q′ forces (in ultralaver forcingLD) that˜Zgk ⊏ r for all k.24

Proof. By (2.26) we can find a sequence (ñk)∞k=1 of L-names such thatq

˜Zgk ⊏

ñk r for eachk. By25

Lemma 2.34(2) we can findq′ ≤0 q be such that this sequence is continuous belowq′. Since eachñk is26

now a front name belowq′, we can apply the previous lemma. 27

Lemma 2.42. Let M be a countable model, r∈ 2ω, DM ∈ M an ultrafilter system,D a filter system28

extendingDM, q ∈ LD. For any V-generic filter G⊆ LD we write GM for the (M-generic, by Lemma 2.4)29

filter onLDM .30

The following are equivalent:31

(1) q LDr is random over M[GM].32

(2) For all names˜Z ∈ M of codes for null sets: q LD ˜

Z ⊏ r.33

(3) For all continuous names g∈ M: q LD ˜Zg ⊏ r.34

Proof. (1)⇔(2) holds because every null set is contained in a set of the form nullset(Z), for some codeZ.35

(2)⇔(3): Every code for a null set inM[GM] is equal to˜Zg[GM], for someg ∈ M, by Corollary 2.35. 36

Lemma 2.43. Let r be random over a countable model M. Then there is a countable model M′ ⊇ M such37

that (2ω)M is countable in M′, but r is still random over M′.38

Proof. We will need the following forcing notions, all defined inM:39

MC

//

B1

MC

˜B2

MB1

˜P=C∗

˜B2/B1

// MC∗˜B2

• Let C be the forcing that collapses the continuum toω with finite conditions.1

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• Let B1 be random forcing (treesT ⊆ 2<ω of positive measure).2

• Let˜B2 be theC-name of random forcing.3

• Let i : B1→ C ∗˜B2 be the natural complete embeddingT 7→ (1C,T).4

• Let˜P be aB1-name for the forcingC∗

˜B2/i[GB1], the quotient ofC∗

˜B2 by the complete subforcing5

i[B1].6

The random realr is B1-generic overM. In M[r] we letP≔˜P[r]. Now let H ⊆ P be generic overM[r].7

Thenr ∗ H ⊆ B1 ∗˜P ≃ C ∗

˜B2 induces anM-generic filterJ ⊆ C and anM[J]-generic filterK ⊆

˜B2[J]; it8

is easy to check thatK interprets the˜B2-name of the canonical random real as the given random realr.9

Hencer is random over the countable modelM′ ≔ M[J], and (2ω)M is countable inM′.10

MJ

//

r

M[J]

K

M[r]H

// M[r][H]

11

Proof of Lemma 2.29.We will first describe a construction that deals with a singletriple (p, ¯˜Z, Z∗) (where12

p is a sequence of conditions with strictly increasing stems which interprets˜Z asZ∗); this construction will13

yield a conditionq′ = q′(p, ¯˜Z, Z∗). We will then show how to deal with all possible triples.14

So letp = p0 be a condition, and let ¯p = (pk)k∈ω be a sequence interpreting˜Z asZ∗, where the lengths15

of the stems ofpn are strictly increasing. It is easy to see that it is enough todeal with a single null set, i.e.,16

m= 1, and withk1 = 0. We write˜Z andZ∗ instead of

˜Z1 andZ∗.17

Using Lemma 2.37 we may (strengthening the conditions in ourinterpretation) assume (inM) that the18

sequence (˜Z(k))k∈ω is almost continuous, witnessed byg : p→ Ω<ω. By Lemma 2.43, we can find a model19

M′ ⊇ M such that (2ω)M is countable inM′, butr is still random overM′.20

We now work inM′. Note thatg still defines an almost continuous name, which we again call˜Z.21

Each filter inDMs is now countably generated; letAs be a pseudo-intersection ofDM

s which additionally22

satisfiesAs ⊆ succp(s) for all s ∈ p above the stem. LetD′s be the Frechet filter onAs. Let p′ ∈ LD′ be the23

tree with the same stem asp which satisfies succp′ (s) = As for all s ∈ p′ above the stem.24

By Lemma 2.4, we know thatLDM is anM-complete subforcing ofLD′ (in M′ as well as inV.) We write25

GM for the induced filter onLDM .26

We now work inV. Note that below the conditionp′, the forcingLD′ is just Laver forcingL, and thatp′ ≤L p. Using Lemma 2.28 we can find a conditionq ≤ p′ (in Laver forcingL) such that:

q is M′-generic.(2.44)

q L r is random overM′[GL] (hence also overM[GM]).(2.45)

Moreover,q L˜Z ⊏0 r.(2.46)

Enumerate all continuousLDM -names for codes for null sets fromM as˜Zg1,

˜Zg2, . . . Applying Corol-27

lary 2.41 yields a conditionq′ ≤ q such that for all filter systemsE satisfyingq′ ∈ LE, we haveq′ LE28

˜Zgi ⊏ r for all i. Corollary 2.39 and Lemma 2.42 now implies:29

(2.47)For every filter systemE satisfyingq′ ∈ LE, q′ forces inLE that r is random overMand that

˜Z ⊏0 r.

By thinning outq′ we may assume that30

(2.48) For eachν ∈ ωω ∩ M there isk such thatνk < q′.

We have now described a construction ofq′ = q′(p,˜Z,Z∗).31

Let (pn,˜Zn,Z∗n) enumerate all triples ( ¯p,

˜Z,Z∗) ∈ M where p interprets

˜Z asZ∗ (and consists of con-32

ditions with strictly increasing stems). For eachn write νn for⋃

k stem(pnk), the branch determined by the33

stems of the sequence ¯pn. We now define by induction a sequenceqn of conditions:34

• q0 = q′(p0,˜Z0,Z∗0).35

• Givenqn−1 and (pn,˜Zn,Z∗n), we findk0 such thatνnk0 < q0 ∪ · · · ∪ qn−1 (using (2.48)). Letk1 be36

such that stem(pnk1

) has length> k0. We replace ¯pn by p′ ≔ (pnk)k≥k1. (Obviously,p′ still interprets37

˜Zn asZ∗n.) Now letqn

≔ q′(p′,˜Zn,Z∗n).1

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Note that the stem ofqn is at least as long as the stem ofpnk1

, and is therefore not inq0 ∪ · · · ∪ qn−1, so2

stem(qi) and stem(q j) are incompatible for alli , j. Therefore we can choose for eachs an ultrafilterDs3

extendingDMs such that stem(qi) ⊆ s implies succqi (s) ∈ Ds.4

Note that allqi are inLD. Therefore, we can use (2.47). Also,qi ≤ pi0.

5

3. J 6

In this section, we define a family of forcing notions that hastwo faces (hence the name “Janus forcing”):7

Elements of this family may be countable (and therefore equivalent to Cohen), and they may also be8

essentially random.9

We will need the following properties of the Janus forcing notionsJ:10

Wish List 3.1. Throughout the whole paper we fix a functionB∗ : ω → ω given by Corollary 3.3. The11

Janus forcings will depend on a real parameterℓ∗ = (ℓ∗m)m∈ω ∈ ωω which grows fast with respect toB∗. (In12

our application,ℓ∗ will be given by a subsequence of an ultralaver real.)13

The sequenceℓ∗ and the functionB∗ together define a notion of a “thin” set, see Definition 2.21.14

(1) There is a canonicalJ-name for a (code for a) null set˜Z∇. WheneverX ⊆ 2ω is not thin, andJ is15

countable, thenJ forces thatX is not strongly meager, witnessed by nullset(˜Z∇). Moreover, for any16

σ-centeredJ-name of a forcing˜Q, alsoJ ∗

˜Q forces thatX is not strongly meager, again witnessed17

by nullset(˜Z∇). (See Definitions 2.21 and 2.24, as well as Lemma 3.9.)18

(2) “Canonical” above in particular means: IfM is a countable model of ZFC*,JM ∈ M is a Janus19

forcing whichM-completely embeds into the Janus forcingJ, then in theJ-extensionV[H],˜Z∇[H]20

is the same as˜ZM∇

[HM], whereHM is the induced filter onJM.21

(3) Let M be a countable transitive model andJM a Janus forcing inM. ThenJM is a Janus forcing in22

V as well (and of course countable inV). (See Fact 3.8.)23

(4) WheneverM is a countable transitive model,JM is a Janus forcing inM, then there is a Janus24

forcing J such that25

• J is (in V) equivalent to random forcing (actually we just need some forcing that strongly26

preserves random reals)27

• JM is anM-complete subforcing ofJ.28

(See Lemma 3.16.)29

(Advice to the reader: At first reading, you may want to take these properties as granted, and skip to30

Section 4.)31

3.A. Definition of Janus. A Janus forcingJ will consist of:32

• A countable “core” (or: backbone)∇which is defined in a combinatorial way from a parameterℓ∗.33

(In our application, we will use a Janus forcing immediatelyafter an ultralaver forcing, andℓ∗ will34

be a subsequence of the ultralaver real.) This core is of course equivalent to Cohen forcing.35

• Some additional “stuffing” J \∇ (countable24 or uncountable). We allow great freedom for this, we36

just require that the core is a “sufficiently” complete subforcing (in a specific combinatorial sense).37

We will use the following combinatorial theorem from [BS10]:38

Lemma 3.2([BS10, Theorem 8]25). For everyε, δ > 0 there exists Nε,δ ∈ ω such that for all sufficiently39

large finite sets I⊆ ω there is a nonempty familyAI consisting of sets A⊆ 2I ,|A|

2|I |≤ ε such that if X⊆ 2I ,40

|X| ≥ Nε,δ then41

|A ∈ AI : X + A = 2I |

|AI |≥ 1− δ.

(Recall that X+ A≔ x+ a : x ∈ X, a ∈ A.)42

Rephrasing and specializing toδ = 14 andε = 1

2i we get:1

24Also the trivial caseJ = ∇ is allowed.25The theorem in [BS10] actually says “for a sufficiently largeI”, but the proof shows that this should be read as “for all sufficiently

largeI”.

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Corollary 3.3. For every i ∈ ω there exists B∗(i) such that for all finite sets I with|I | ≥ B∗(i) there is a2

nonempty familyAI satisfying the following:3

• AI consists of sets A⊆ 2I with|A|

2|I |≤

12i

.4

• For every X⊆ 2I satisfying|X| ≥ B∗(i), the setA ∈ AI : X + A = 2I has at least34 |AI | elements.5

Assumption 3.4. We fix a sufficiently fast increasing sequenceℓ∗ = (ℓ∗i )i∈ω of natural numbers; more6

precisely, the sequenceℓ∗ will be a subsequence of an ultralaver realℓ, defined as in Lemma 2.22 using the7

functionB∗ from Corollary 3.3. Note that in this caseℓ∗i+1 − ℓ∗i ≥ B∗(i); so we can fix for eachi a family8

Ai ⊆P(2Li ) on the intervalLi ≔ [ℓ∗i , ℓ∗i+1) according to Corollary 3.3.

9

Definition 3.5. First we define the “core”∇ = ∇ℓ∗ of our forcing:10

∇ =⋃

i∈ω

∏

j<i

A j .

In other words,σ ∈ ∇ iff σ = (A0, . . . ,Ai−1) for somei ∈ ω, A0 ∈ A0, . . . , Ai−1 ∈ Ai−1. We will denote the11

numberi by height(σ).12

The forcing notion∇ is ordered by reverse inclusion (i.e., end extension):τ ≤ σ if τ ⊇ σ.13

Definition 3.6. Let ℓ∗ = (ℓ∗i )i∈ω be as in the assumption above. We say thatJ is a Janus forcing based onℓ∗14

if:15

(1) (∇,⊇) is an incompatibility-preserving subforcing ofJ.16

(2) For eachi ∈ ω the setσ ∈ ∇ : height(σ) = i is predense inJ. So in particular,J adds a17

branch through∇. The union of this branch is called˜C∇ = (

˜C∇0 , ˜

C∇1 , ˜C∇2 , . . .), where

˜C∇i ⊆ 2Li with18

˜C∇i ∈ Ai .19

(3) “Fatness”:26 For all p ∈ J and all real numbersε > 0 there are arbitrarily largei ∈ ω such that there20

is a core conditionσ = (A0, . . . ,Ai−1) ∈ ∇ (of lengthi) with21

|A ∈ Ai : σA ‖J p ||Ai |

≥ 1− ε.

(4) J is ccc.22

(5) J is separative.23

(6) (To simplify some technicalities:)J ⊆ H(ℵ1).24

We now define˜Z∇, which will be a canonicalJ-name for (a code for) a null set. We will use the sequence25

˜C∇ added byJ (see Definition 3.6(2)).26

Definition 3.7. Each˜C∇i defines a clopen set

˜Z∇i = x ∈ 2ω : xLi ∈

˜C∇i of measure at most12i+1 . The27

sequence˜Z∇ = (

˜Z∇0 , ˜

Z∇1 , ˜Z∇2 , . . .) is (a name for) a code for the null set28

nullset(˜Z∇) =

⋂

n<ω

⋃

i≥n˜Z∇i .

For simplicity, we will write˜Z∇ instead of nullset(

˜Z∇).29

For later reference, we record the following trivial fact:30

Fact 3.8. Let (Mn)n∈ω be an increasing sequence of countable models, and letJn ∈ Mn be Janus forcings.31

Assume thatJn is Mn-complete inJn+1. Then⋃

n Jn is a Janus forcing, and anMn-complete extension of32

Jn for all n.33

This is also true if allMn are equal. In other words: IfJ ∈ M is a Janus forcing inM, thenJ is also a34

Janus forcing inV.1

26Actually, (3) implies (2).

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3.B. Janus and strongly meager.Carlson [Car93] showed that Cohen reals make every uncountable2

set X of the ground model not strongly meager in the extension (andthat not being strongly meager is3

preserved in a subsequent forcing with precaliberℵ1). We show that acountableJanus forcingJ does the4

same (for a subsequent forcing that is evenσ-centered, not just precaliberℵ1). This sounds trivial, since5

any (nontrivial) countable forcing is equivalent to Cohen forcing anyway. However, we show (and will6

later use) that the canonical null set˜Z∇ defined above witnesses thatX is not strongly meager (and not7

just some null set that we get out of the isomorphism betweenJ and Cohen forcing). The point is that8

while∇ is not a complete subforcing ofJ, the condition (3) of the Definition 3.6 guarantees that Carlson’s9

argument still works, if we assume thatX is non-thin (not just uncountable). This is enough for us, since10

by Corollary 2.23 ultralaver forcing makes any uncountableset non-thin.11

Recall that we fixed the increasing sequenceℓ∗ = (ℓ∗i )i∈ω andB∗. In the following, whenever we say12

“(very) thin” we mean “(very) thin with respect toℓ∗ andB∗” (see Definition 2.21).13

Lemma 3.9. If X is not thin,J is a countable Janus forcing based onℓ∗, and˜R is aJ-name for aσ-centered14

forcing notion, thenJ ∗˜R forces that X is not strongly meager witnessed by the null set

˜Z∇.15

Proof. Let˜c be aJ-name for a function

˜c :

˜R→ ω witnessing that

˜R isσ-centered.16

Recall that “˜Z∇ witnesses thatX is not strongly meager” means thatX +

˜Z∇ = 2ω. Assume towards17

a contradiction that (p, r) ∈ J ∗˜R forces thatX +

˜Z∇ , 2ω. Then we can fix a (J ∗

˜R)-name

˜ξ such that18

(p, r) ˜ξ < X +

˜Z∇, i.e., (p, r) (∀x ∈ X)

˜ξ < x+

˜Z∇. By definition of

˜Z∇, we get19

(p, r) (∀x ∈ X) (∃n ∈ ω) (∀i ≥ n)˜ξLi < xLi +

˜C∇i .

For eachx ∈ X we can find (px, rx) ≤ (p, r) and natural numbersnx ∈ ω andmx ∈ ω such thatpx forces20

that˜c(rx) = mx and21

(px, rx) (∀i ≥ nx)˜ξLi < xLi +

˜C∇i .

SoX =⋃

p∈J,m∈ω,n∈ω Xp,m,n, whereXp,m,n is the set of allx with px = p, mx = m, nx = n. (Note thatJ is22

countable, so the union is countable.) AsX is not thin, there is somep∗,m∗, n∗ such thatX∗ ≔ Xp∗ ,m∗ ,n∗ is23

not very thin. So we get for allx ∈ X∗:24

(3.10) (p∗, rx) (∀i ≥ n∗)˜ξLi < xLi +

˜C∇i .

SinceX∗ is not very thin, there is somei0 ∈ ω such that for alli ≥ i025

(3.11) the (finite) setX∗Li has more thanB∗(i) elements.

Due to the fact thatJ is a Janus forcing (see Definition 3.6 (3)), there are arbitrarily large i ∈ ω such that26

there is a core conditionσ = (A0, . . . ,Ai−1) ∈ ∇ with27

(3.12)|A ∈ Ai : σA ‖J p∗|

|Ai |≥

23.

Fix such an i larger than bothi0 andn∗, and fix a conditionσ satisfying (3.12).28

We now consider the following two subsets ofAi :29

(3.13) A ∈ Ai : σA ‖J p∗ and A ∈ Ai : X∗Li + A = 2Li .

By (3.12), the relative measure (inAi) of the left one is at least23; due to (3.11) and the definition ofAi30

according to Corollary 3.3, the relative measure of the right one is at least34; so the two sets in (3.13) are31

not disjoint, and we can pick anA belonging to both.32

Clearly,σA forces (inJ) that˜C∇i is equal toA. Fix q ∈ J witnessingσA ‖J p∗. Then33

(3.14) q J X∗Li +˜C∇i = X∗Li + A = 2Li .

Sincep∗ forces that for eachx ∈ X∗ the color˜c(rx) = m∗, we can find anr∗ which is (forced byq ≤ p∗34

to be) a lower bound of thefinitesetrx : x ∈ X∗∗, whereX∗∗ ⊆ X∗ is any finite set withX∗∗Li = X∗Li .35

By (3.10),36

(q, r∗) ˜ξLi < X∗∗Li +

˜C∇i = X∗Li +

˜C∇i ,

contradicting (3.14). 1

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Corollary 3.15. Let X be uncountable. IfLD is any ultralaver forcing adding an ultralaver realℓ, and ℓ∗2

is defined fromℓ as in Lemma 2.22, and if˜J is a countable Janus forcing based onℓ∗,

˜Q is anyσ-centered3

forcing, thenLD ∗ ˜J ∗

˜Q forces that X is not strongly meager.

4

3.C. Janus may be random.We show here that every countable Janus forcing can be embedded into a5

Janus forcing which is equivalent to random forcing.6

Lemma 3.16. Let M be a countable model of ZFC*, and letJM ∈ M be a Janus forcing (based onℓ∗).7

Then there is a Janus forcingJ (based onℓ∗) such that8

• JM ⋖M J (i.e.,JM is an M-complete subforcing ofJ).9

• J is forcing equivalent to random forcing.10

Proof. Let rn : n ∈ ω be an enumeration ofJM ≔ JM. Let Dk : k ∈ ω be an enumeration of allD ∈ M11

which are open dense subsets ofJM. Recall that∇ = ∇JM

was defined in Definition 3.5.12

We now fixn for a while (up to (3.18)). We will construct a finitely splitting treeSn ⊆ ω<ω and a family13

(σns, p

ns, τ∗ns )s∈Sn satisfying the following (suppressing the superscriptn):14

(a) σs ∈ ∇, σ〈〉 = 〈〉, s⊆ t impliesσs ⊆ σt, ands⊥Sn t impliesσs ⊥∇ σt.15

(So in particular the setσt : t ∈ succSn(s) is a (finite) antichain aboveσs in ∇.)16

(b) ps ∈ JM, p〈〉 = rn; if s⊆ t thenpt ≤JM ps (hencept ≤ rn); s⊥Sn t implies ps ⊥JM pt.17

(c) ps ≤JM σs.18

(d) σs ⊆ τ∗s ∈ ∇, andσt : t ∈ succSn(s) is the set of allτ ∈ succ∇(τ∗s) which are compatible withps.19

(e) The setσt : t ∈ succSn(s) is a subset of succ∇(τ∗s) of relative size at least 1− 1lh(s)+10.

20(f) Eachs ∈ Sn has at least 2 successors (inSn).21

(g) If k = lh(s), thenps ∈ Dk (and therefore also in allDl for l < k).22

Setσ〈〉 = 〈〉 and p〈〉 = rn. Given s, σs and ps, we construct succSn(s) and (σt, pt)t∈succSn (s): We apply23

fatness 3.6(3) tops with ε = 1lh(s)+10. So we get someτ∗s ∈ ∇ of height bigger than the height ofσs such24

that the setB of elements of succ∇(τ∗s) which are compatible withps has relative size at least 1− ε. Since25

ps ≤JM σs we get thatτ∗s is compatible with (and therefore stronger than)σs. EnumerateB asτ0, . . . , τl−1.26

Set succSn(s) = si : i < l andσs i = τi . For t ∈ succS(s), choosept ∈ JM stronger than bothσt andps27

(which is obviously possible sinceσt andps are compatible), and moreoverpt ∈ Dlh(t). This concludes the28

construction of the family (σns, p

ns, τ∗ns )s∈Sn.29

So (Sn,⊆) is a finitely splitting nonempty tree of heightω with no maximal nodes and no isolated30

branches. [Sn] is the (compact) set of branches ofSn. The closed subsets of [Sn] are exactly the sets of31

the form [T], whereT ⊆ Sn is a subtree ofSn with no maximal nodes. [Sn] carries a natural (“uniform”)32

probability measureµn, which is characterized by33

µn((Sn)[t]) =1

|succSn(s)|· µn((Sn)[s])

for all s ∈ Sn and allt ∈ succSn(s). (We just writeµn(T) instead ofµn([T]) to increase readability.)34

We callT ⊆ Sn positive if µn(T) > 0, and we callT pruned ifµ(T [s]) > 0 for all s ∈ T. (Clearly every35

treeT contains a pruned treeT′ of the same measure, which can be obtained fromT by removing all nodes36

swith µ(T [s]) = 0.)37

Let T ⊆ Sn be a positive, pruned tree andε > 0. Then on all but finitely many levelsk there is ans ∈ T38

such that39

(3.17) succT (s) ⊆ succSn(s) has relative size≥ 1− ε.

(This follows from Lebesgue’s density theorem, or can easily be seen directly: SetCm =⋃

t∈T, lh(t)=m (Sn)[t] .40

ThenCm is a decreasing sequence of closed sets, each containing [T]. If the claim fails, thenµn(Cm+1)) ≤41

µn(Cm) · (1− ε) infinitely often; soµn(T) ≤ µ(⋂

mCm) = 0.)42

It is well known that the set of positive, pruned subtrees ofSn, ordered by inclusion, is forcing equivalent43

to random forcing (which can be defined as the set of positive,pruned subtrees of 2<ω).44

We have now constructedSn for all n. Define

J = JM ∪⋃

n

(n,T) : T ⊆ Sn is a positive pruned tree

(3.18)

with the following partial order:1

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• The order onJ extends the order onJM.2

• (n′,T′) ≤ (n,T) if n = n′ andT′ ⊆ T.3

• (n,T) ≤ p if there is ak such thatpnt ≤ p for all t ∈ T of lengthk. (Note that this will then be true4

for all biggerk as well.)5

• p ≤ (n,T) never holds.6

The lemma now easily follows from the following properties:7

(1) The order onJ is transitive.8

(2) JM is an incompatibility-preserving subforcing ofJ.9

In particular,J satisfies item (1) of Definition 3.6 of Janus forcing.10

(3) For allk: the set(n,T [t]) : t ∈ T, lh(t) = k is a (finite) predense antichain below (n,T).11

(4) (n,T [t]) is stronger thanpnt for eacht ∈ T (witnessed, e.g., byk = lh(t)). Of course, (n,T [t]) is12

stronger than (n,T) as well.13

(5) Sincepnt ∈ Dk for k = lh(t), this implies that eachDk is predense below each (n,Sn) and therefore14

in J.15

So in particular,JM ⋖M J, i.e.,JM is anM-complete subforcing ofJ.16

Also, for eachk the setσ ∈ ∇ : height(σ) = k is in M (and predense inJM). Therefore this set is17

predense inJ as well, i.e., item (2) of the Definition 3.6 of Janus forcing is satisfied.18

(6) The condition (n,Sn) is stronger thanrn, so (n,Sn) : n ∈ ω is predense inJ andJ \ JM is dense19

in J.20

Below each (n,Sn), the forcingJ is isomorphic to random forcing.21

Therefore,J itself is forcing equivalent to random forcing. (In fact, the complete Boolean algebra22

generated byJ is isomorphic to the standard random algebra, Borel sets modulo null sets.)23

(7) The remaining item of the definition of Janus forcing, fatness 3.6(3), is satisfied.24

I.e., given (n,T) ∈ J andε > 0 there is an arbitrarily highτ∗ ∈ ∇ such that the relative size of the25

setτ ∈ succ∇(τ∗) : τ ‖ (n,T) is at least 1− ε. (We will show≥ (1− ε)2 instead, to simplify the26

notation.)27

We show (7): Given (n,T) ∈ J andε > 0, we use (3.17) to get an arbitrarily highs ∈ T such that succT(s)28

is of relative size≥ 1− ε in succS(s). We may choosesof length> 1ε. We claim thatτ∗s is as required:

29

• Let B≔ σt : t ∈ succS(s). Note thatB = τ ∈ succ∇(τ∗s) : τ ‖ ps.30

B has relative size≥ 1− 1lh(s) ≥ 1− ε in succ∇(τ∗s) (according to property (e) ofSn).

31• C ≔ σt : t ∈ succT(s) is a subset ofB of relative size≥ 1− ε according to our choice ofs.32

• SoC is of relative size (1− ε)2 in succ∇(τ∗s).33

• Eachσt ∈ C is compatible with (n,T), as (n,T [t]) ≤ pt ≤ σt (see (4)).34

It is easy (but not even necessary) to check thatJ is separative. In any case, we could replace≤J by ≤∗J,35

thus makingJ separative without changing≤JM , sinceJM was already separative. 36

4. A 37

A main tool to construct the forcing for BC+dBC will be “partial countable support iterations”, more38

particularly “almost finite support” and “almost countablesupport” iterations. A partial countable support39

iteration is a forcing iteration (Pα,Qα)α<ω2 such that for each limit ordinalδ the forcing notionPδ is a subset40

of the countable support limit of (Pα,Qα)α<δ which satisfies some natural properties (see Definition 4.6).41

Instead of transitive models, we will use ord-transitive models (which are transitive when ordinals are42

considered as urelements). Why do we do that? We want to “approximate” the generic iterationP of length43

ω2 with countable models; this can be done more naturally with ord-transitive models (since obviously44

countable transitive models only see countable ordinals).We call such an ord-transitive model a “candi-45

date” (provided it satisfies some nice properties, see Definition 4.2). A basic point is that forcing extensions46

work naturally with candidates.47

In the following,x = (Mx, Px) will denote a pair such thatMx is a candidate andPx is (in Mx) a partial48

countable support iteration; similarly we write, e.g.,y = (My, Py) or xn = (Mxn, Pxn).49

We will need the following results to prove BC+dBC. (However, other than in the case of the ultralaver50

and Janus section, the reader will probably have to read thissection to understand the construction in the51

next section, and not just the wish list.)1

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Wish List 4.1. Givenx = (Mx, Px), we can construct by induction onα a partial countable support iteration2

P = (Pα,Qα)α<ω2 satisfying:3

There is a canonicalMx-complete embedding fromPx to P.4

In this construction, we can use at each stageβ any desiredQβ, as long asPβ forces thatQxβ

is (evaluated5

as) anMx[Hxβ]-complete subforcing ofQβ (whereHx

β⊆ Px

βis theMx-generic filter induced by the generic6

filter Hβ ⊆ Pβ).7

Moreover, we can demand either of the following two additional properties of this limit:278

(1) If all Qβ are forced to beσ-centered, andQβ is trivial for all β < Mx, thenPω2 isσ-centered.9

(2) If r is random overMx, and allQβ locally preserve randomness ofr over Mx[Hxβ] (see Defini-10

tion 4.31), then alsoPω2 locally preserves the randomness ofr.11

Actually, we need the following variant: Assume that we already havePα0 for someα0 ∈ Mx, and thatPxα0

12

canonically embeds intoPα0, and that the respective assumption onQβ holds for allβ ≥ α0. Then we get13

thatPα0 forces that the quotientPω2/Pα0 satisfies the respective conclusion.14

We also need:2815

(3) If instead of a singlex we have a sequencexn such that eachPxn canonically (andMxn-completely)16

embeds intoPxn+1, then we can find a partial countable support iterationP into which allPxn embed17

canonically (and we can again use any desiredQβ, assuming thatQxnβ

is an Mxn[Hxnβ

]-complete18

subforcing ofQβ for all n ∈ ω).19

(4) (A fact that is easy to prove but awkward to formulate.) Ifa ∆-system argument produces two20

x1, x2 as in Lemma 5.7(3), then we can find a partial countable support iterationP such thatPxi21

canonically (andMxi -completely) embeds intoP for i = 1, 2.22

4.A. Ord-transitive models. We will use “ord-transitive” models, as introduced in [She04] (see also the23

presentation in [Kel]). We briefly summarize the basic definitions and properties (restricted to the rather24

simple case needed in this paper):25

Definition 4.2. Fix a suitable finite subset ZFC∗ of ZFC (that is satisfied byH(χ∗) for sufficiently large26

regularχ∗).27

(1) A setM is calledcandidate, if28

• M is countable,29

• (M, ∈) is a model of ZFC∗,30

• M is ord-absolute:M |= α ∈ Ord iff α ∈ Ord, for allα ∈ M,31

• M is ord-transitive: ifx ∈ M \Ord, thenx ⊆ M,32

• ω + 1 ⊆ M.33

• “α is a limit ordinal” and “α = β + 1” are both absolute betweenM andV.34

(2) A candidateM is callednice, if “ α has countable cofinality” and “the countable setA is cofinal35

in α” both are absolute betweenM andV. (So if α ∈ M has countable cofinality, thenα ∩ M is36

cofinal inα.) Moreover, we assumeω1 ∈ M (which impliesω1M = ω1) andω2 ∈ M (but we do37

not requireω2M = ω2).38

(3) Let PM be a forcing notion in a candidateM. (To simplify notation, we can assume without loss39

of generality thatPM ∩ Ord = ∅ (or at least⊆ ω) and that thereforePM ⊆ M and alsoA ⊆ M40

wheneverM thinks thatA is a subset ofPM.) Recall that a subsetHM of PM is M-generic (or:41

PM-generic overM), if |A∩ HM | = 1 for all maximal antichainsA in M.42

(4) LetHM bePM-generic overM and˜τ a PM-name inM. We define the evaluation

˜τ[HM]M to bex if43

M thinks thatp PM

˜τ = x for somep ∈ HM andx ∈ M (or equivalently just forx ∈ M∩Ord), and44

˜σ[HM]M : (

˜σ, p) ∈

˜τ, p ∈ HM otherwise. Abusing notation we write

˜τ[HM] instead of

˜τ[HM]M,45

and we writeM[HM] for ˜τ[HM] :

˜τ is aPM-name inM.46

(5) The ord-collapsek (or kM) is a recursively defined function with domainM: k(x) = x if x ∈ Ord,47

andk(x) = k(y) : y ∈ x∩ M otherwise.1

27Theσ-centered version is central for the proof of dBC; the randompreserving version for BC.28This will give σ-closure andℵ2-cc for the preparatory forcingR.

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(6) The ord-transitive closure of a setx is defined inductively on the rank:2

ordclos(x) = x∪⋃

ordclos(y) : y ∈ x \Ord.

So ordclos(x) is the smallest ord-transitive set containingx as a subset. HCON is the collection of3

all setsx such that the ord-transitive closure ofx is countable.x is in HCON iff x is element of4

some candidate. In particular, all reals and all ordinals are HCON.5

We write HCONα for the family of all setsx in HCON whose ord-transitive closure (or, in this6

case equivalently, transitive closure) only contains ordinals< α.7

Fact 4.3. (1) The ord-collapse of a countable elementary submodel ofH(χ∗) is a nice candidate.8

(2) Unions, intersections etc. are generally not absolute for candidates. For example, letx ∈ M \Ord.9

In M we can construct a sety such thatM |= y = ω1 ∪ x. Theny is not an ordinal and therefore a10

subset ofM, and in particulary is countable andy , ω1 ∪ x.11

(3) Let j : M → M′ be the transitive collapse of a candidateM, and f : ω1 ∩ M′ → Ord the inverse12

(restricted to the ordinals). ObviouslyM′ is a countable transitive model of ZFC∗; moreoverM13

is characterized by the pair (M′, f ) (we call such a pair a “labeled transitive model”). Note that f14

satisfiesf (α + 1) = f (α) + 1, f (α) = α for α ∈ ω ∪ ω. M |= (α is a limit) iff f (α) is a limit.15

M |= cf(α) = ω iff cf( f (α)) = ω, and in that casef [α] is cofinal inα. On the other hand, given16

a transitive countable modelM′ of ZFC∗ and an f as above, then we can construct a (unique)17

candidateM corresponding to (M′, f ).18

(4) All candidatesM with M ∩ Ord ⊆ ω1 are hereditarily countable, so their number is at most 2ℵ0.19

Similarly, the cardinality of HCONα is at most continuum wheneverα < ω2.20

(5) If HM is PM-generic overM, thenM[HM] is a candidate as well and an end-extension ofM such21

thatM ∩Ord= M[HM] ∩Ord. If M is nice and (M thinks that)PM is proper, thenM[HM] is nice22

as well.23

(6) Forcing extensions commute with the transitive collapse j: If M corresponds to (M′, f ), then24

HM ⊆ PM is PM-generic overM iff H′ ≔ j[HM] is P′ ≔ j(PM)-generic overM′, and in that case25

M[HM] corresponds to (M′[H′], f ). In particular, the forcing extension ofM[HM] of M satisfies26

the forcing theorem (everything that is forced is true, and everything true is forced).27

(7) For elementary submodels, forcing extensions commute with ord-collapses: LetN be a countable28

elementary submodel ofH(χ∗), P ∈ N, k : N → M the ord-collapse (soM is a candidate), and let29

H beP-generic overV. ThenH is P-generic overN iff HM≔ k[H] is PM

≔ k(P)-generic overM;30

and in that case the ord-collapse ofN[H] is M[HM].31

Assume that a nice candidateM thinks that (PM, QM) is a forcing iteration of lengthω2V (we will32

usually writeω2 for the length of the iteration, by this we will always meanω2V and not the possibly33

differentω2M). In this section, we will construct an iteration (P, Q) in V, also of lengthω2, such that each34

PMα canonically andM-completely embeds intoPα for all α ∈ ω2 ∩ M. Once we know (by induction) that35

PMα M-completely embeds intoPα, we know that aPα-generic filterHα induces aPM

α -generic (overM)36

filter which we callHMα . ThenM[HM

α ] is a candidate, but nice only ifPMα is proper. We will not need that37

M[HMα ] is nice, actually we will only investigate set of reals (or elements ofH(ℵ1)) in M[HM

α ], so it does38

not make any difference whether we useM[HMα ] or its transitive collapse.39

Remark 4.4. In the discussion so far we omitted some details regarding the theory ZFC∗ (that a candidate40

has to satisfy). The following “fine print” hopefully absolves us from any liability. (It is entirely irrelevant41

for the understanding of the paper.)42

We have to guarantee that eachM[HMα ] that we consider satisfies enough of ZFC to make our arguments43

work (for example, the definitions and basic properties of ultralaver and Janus forcings should work). This44

turns out to be easy, since (as usual) we do not need the full power set axiom for these arguments (just the45

existence of, say,i5). So it is enough that eachM[HMα ] satisfies some fixed finite subset of ZFC minus46

power set, which we call ZFC∗.47

Of course we can also find a bigger (still finite) set ZFC∗∗ that implies:i10 exists, and each forcing48

extension of the universe with a forcing of size≤ i4 satisfies ZFC∗. And it is provable (in ZFC) that each49

H(χ) satisfies ZFC∗∗ for sufficiently large regularχ.50

We define candidate using the weaker theory ZFC∗, and require that nice candidates satisfies the stronger51

theory ZFC∗∗. This guarantees that all forcing extensions (by small forcings) of nice candidates will be1

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candidates (in particular, satisfy enough of ZFC such that our arguments about Janus or ultralaver forcings2

work). Also, every ord-collapse of a countable elementary submodelN of H(χ) will be a nice candidate.3

4.B. Partial countable support iterations. We introduce the notion of “partial countable support limit”:4

a subset of the countable support (CS) limit containing the union (i.e., the direct limit) and satisfying some5

natural requirements.6

Let us first describe what we mean by “forcing iteration”. They have to satisfy the following require-7

ments:8

• A “ topless forcing iteration” ( Pα,Qα)α<ε is a sequence of forcing notionsPα andPα-namesQα9

of quasiorders with a weakest element 1Qα . A “ topped iteration” additionally has a final limitPε.10

EachPα is a set of partial functions onα (as, e.g., in [Gol93]). More specifically, ifα < β ≤ ε11

and p ∈ Pβ, thenpα ∈ Pα. Also, pβ Pβ p(β) ∈ Qβ for all β ∈ dom(p). The order onPβ will12

always be the “natural” one:q ≤ p iff qα forces (inPα) thatqtot(α) ≤ ptot(α) for all α < β, where13

r tot(α) = r(α) for all α ∈ dom(r) and 1Qα otherwise.Pα+1 consists ofall p with pα ∈ Pα and14

pα ptot(α) ∈ Qα, so it is forcing equivalent toPα ∗ Qα.15

• Pα ⊆ Pβ wheneverα < β ≤ ε. (In particular, the empty condition is an element of eachPβ.)16

• For anyp ∈ Pε and anyq ∈ Pα (α < ε) with q ≤ pα, the partial functionq∧ p≔ q∪ p[α, ε) is a17

condition inPε as well (so in particular,pα is a reduction ofp, hencePα is a complete subforcing18

of Pε; andq∧ p is the weakest condition inPε stronger than bothq andp).19

• Abusing notation, we usually just writeP for an iteration (be it topless or topped).20

• We usually writeHβ for the generic filter onPβ (which inducesPα-generic filters calledHα for21

α ≤ β). For topped iterations we call the filter on the final limit sometimes justH instead ofHε.22

We use the following notation for factorization of iterations:23

• For α < β, in the Pα-extensionV[Hα], we let Pβ/Hα be the set of allp ∈ Pβ with pα ∈ Hα24

(ordered as inPβ). We may occasionally writePβ/Pα for thePα-name ofPβ/Hα.25

• SincePα is a complete subforcing ofPβ, this is a factorization with the usual properties, in partic-26

ular Pβ is equivalent toPα ∗ (Pβ/Hα).27

Definition 4.5. Let P be a (topless) iteration of limit lengthε. We define three limits ofP:28

• The “direct limit” is the union of thePα (for α < ε). So this is the smallest possible limit of the29

iteration.30

• The “inverse limit” consists ofall partial functionsp with domain⊆ ε such thatpα ∈ Pα for all31

α < ε. This is the largest possible limit of the iteration.32

• The “full countable support limit PCSε ” of P is the inverse limit if cf(ε) = ω and the direct limit33

otherwise.34

We say thatPε is a “partial CS limit”, if Pε is a subset of the full CS limit and the sequence (Pα)α≤ε is a35

topped iteration. In particular, this means thatPε contains the direct limit, and satisfies the following for36

eachα < ε: Pε is closed underp 7→ pα, and wheneverp ∈ Pε, q ∈ Pα, q ≤ pα, then also the partial37

functionq∧ p is in Pε.38

So for a given toplessP there is a well-defined inverse, direct and full CS limit. If cf(ε) > ω, then they39

all coincide. If cf(ε) = ω, then the direct limit and the full CS limit (=inverse limit) differ. Both of them40

are partial CS limits, but there are many more possibilitiesfor partial CS limits. By definition, all of them41

will yield iterations.42

Note that the name “CS limit” is slightly inappropriate, as the size of supports of condition is not part of43

the definition. To give a more specific example: Consider a topped iterationP of lengthω +ω wherePω is44

the direct limit andPω+ω is the full CS limit. Letp be any element of the full CS limit ofPω which is not45

in Pω; thenp is not inPω+ω either. So not every countable subset ofω + ω can appear as the support of a46

condition.47

Definition 4.6. A forcing iterationP is called a “partial CS iteration”, if48

• every limit is a partial CS limit, and49

• everyQα is (forced to be) separative29, i.e.,≤Qα coincides with≤∗Qα .1

29The reason for this requirement is briefly discussed in Section 7

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The following fact can easily be proved by transfinite induction:2

Fact 4.7. Let P be a partial CS iteration. Then for allα the forcing notionPα is separative.3

From now on, all iterations we consider will be partial CS iterations. In this paper, we will only be4

interested in proper partial CS iterations, but propernessis not part of the definition of partial CS iteration.5

(The reader may safely assume that all iterations are proper.)6

Note that separativity of theQα implies that all partial CS iterations satisfy the following (trivially7

equivalent) properties:8

Fact 4.8. Let P be a topped partial CS iteration of lengthε. Then:9

(1) Let H bePε-generic. Thenp ∈ H iff pα ∈ Hα for all α < ε.10

(2) For allq, p ∈ Pε: If qα ≤∗ pα for eachα < ε, thenq ≤∗ p.11

(3) For allq, p ∈ Pε: If qα ≤∗ pα for eachα < ε, thenq ‖ p.12

We will be concerned with the following situation:13

Assume thatM is a nice candidate,PM is (in M) a topped partial CS iteration of lengthε (a limit ordinal14

in M), andP is (in V) a topless partial CS iteration of lengthε′ ≔ sup(ε ∩ M). (Recall that “cf(ε) = ω”15

is absolute betweenM andV, and that cf(ε) = ω impliesε′ = ε.) Moreover, assume that we already have16

a system ofM-complete coherent30 embeddingsiβ : PMβ→ Pβ for β ∈ ε′ ∩ M = ε ∩ M. (Recall that any17

potential partial CS limit ofP is a subforcing of the full CS limitPCSε′ .) It is easy to see that there is only one18

possibility for an embeddingj : PMε → PCS

ε′(in fact, into any potential partial CS limit ofP) that extends19

the iβ’s naturally:20

Definition 4.9. For a topped partial CS iterationPM in M of lengthε and a topless oneP in V of length21

ε′ ≔ sup(ε∩M) together with coherent embeddingsiβ, we definej : PMε → PCS

ε′, the “canonical extension”,22

in the obvious way: Givenp ∈ PMε , take the sequence of restrictions toM-ordinals, apply the functionsiβ,23

and build j(p) from the resulting coherent sequence.24

We do not claim thatj : PMε → PCS

ε′is M-complete.31 In the following, we will construct partial CS25

limits Pε′ such thatj : PMε → Pε′ is M-complete. (Obviously, one requirement for such a limit is that26

j[PMε ] ⊆ Pε′ .) We will actually define two versions: The almost FS and the almost CS limit.27

Instead of arbitrary systems of embeddingsiα, we will only be interested in “canonical” ones. We as-28

sume for notational convenience thatQMα is a subset ofQα (this will naturally be the case in our application29

anyway).30

Definition 4.10 (The canonical embedding). Let P be a partial CS iteration inV and PM a partial CS31

iteration in M, both topped and of lengthε ∈ M. We construct by induction onα ∈ (ε + 1) ∩ M the32

canonicalM-complete embeddingsiα : PMα → Pα. More precisely: We try to construct them, but it33

is possible that the construction fails. If the construction succeeds, then we say that “PM (canonically)34

embeds intoP”, or “ the canonical embeddings work”, or just: “P is overPM”, or “ over PMε ”.35

• Let α = β + 1. By induction hypothesis,iβ is M-complete, so aV-generic filterHβ ⊆ Pβ induces36

anM-generic filterHMβ≔ i−1

β[Hβ] ⊆ PM

β. We require that (in theHβ extension) the setQM

β[HMβ

] is37

anM[HMβ

]-complete subforcing ofQβ[Hβ]. In this case, we defineiα in the obvious way.38

• For α limit, let iα be the canonical extension of the family (iβ)β∈α∩M. (If α′ ≔ sup(α ∩ M) < α,39

theniα is a map intoPα′ , which is a complete subforcing ofPα.) We require thatPα contains the40

range ofiα, and thatiα is M-complete; otherwise the construction fails.41

In this section we try to construct a partial CS iterationP (over a givenPM) satisfying additional prop-42

erties.1

30I.e., they commute with the restriction maps:iα(pα) = iβ(p)α for α < β andp ∈ PMβ

.31 For example, ifε = ε′ = ω and if PM

ω is the finite support limit of a nontrivial iteration, thenj : PMω → PCS

ω is not complete: InM, let cn be (aPM

n -name for) a “nontrivial” element ofQMn (i.e., 1QM

n∗ cn). Let pn be thePM

n -condition “cm ∈ H(m) for all m< n”

and letqn be thePMn+1-condition (c0, . . . , cn−1,¬cn) = pn ∧ ¬pn+1, i.e., “n is minimal withcn < H(n)”. In M, the setA = qn : n ∈ ω

is a maximal antichain inPMω . Moreover, the sequence (pn)n∈ω is a decreasing coherent sequence, thereforein(pn) defines an element

pω in PCSω , which is clearly incompatible with allj(qn), hencej[A] is not maximal.

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Remark 4.11. What is the role ofε′ ≔ sup(ε ∩ M)? When our inductive construction ofP arrives at2

Pε whereε′ < ε, it would be too late32 to take care ofM-completeness ofiε at this stage, even if all3

iα work nicely forα ∈ ε ∩ M. Note thatε′ < ε implies thatε is uncountable inM, and that therefore4

PMε =⋃

α∈ε∩M PMα . So the natural extensionj of the embeddings (iα)α∈ε∩M has range inPε′ , which will be5

a complete subforcing ofPε. So we have to ensureM-completeness already in the construction ofPε′ .6

For now we just record:7

Lemma 4.12. Assume that we have topped iterationsPM (in M) of lengthε andP (in V) of lengthε′ ≔8

sup(ε ∩ M), and that for allα ∈ ε ∩ M the canonical embedding iα : PMα → Pα works. Let iε : PM

ε → PCSε′

9

be the canonical extension.10

(1) If PMε is (in M) a direct limit (which is always the case ifε has uncountable cofinality) then iε11

(might not work, but at least) has range in Pε′ and preserves incompatibility.12

(2) If iε has a range contained in Pε′ and maps predense sets D⊆ PMε in M to predense sets iε[D] ⊆13

Pε′ , then iε preserves incompatibility (and therefore works).14

Proof. (1) SincePMε is a direct limit, the canonical extensioniε has range in

⋃

α<ε′ Pα, which is subset of15

any partial CS limitPε′ . Incompatibility inPMε is the same as incompatibility inPM

α for sufficiently large16

α ∈ ε ∩ M, so it by assumption it is preserved byiα and hence also byiε.17

(2) Fix p1, p2 ∈ PMε , and assume that their images are compatible inPε′ ; we have to show that they are18

compatible inPMε . So fix a generic filterH ⊆ Pε′ containingiε(p1) andiε(p2).19

In M, we define the following setD:20

D ≔ q ∈ PMε : (q ≤ p1 ∧ q ≤ p2) or (∃α < ε : qα ⊥PM

αp1α) or (∃α < ε : qα ⊥PM

αp2α).

Using Fact 4.8(3) it is easy to check thatD is dense. Sinceiε preserves predensity, there isq ∈ D21

such thatiε(q) ∈ H. We claim thatq is stronger thanp1 and p2. Otherwise we would have without loss22

of generalityqα ⊥PMα

p1α for someα < ε. But the filterHα contains bothiα(qα) and iα(p1α),23

contradicting the assumption thatiα preserves incompatibility. 24

4.C. Almost finite support iterations. Recall Definition 4.9 (of the canonical extension) and the setup25

that was described there: We have to find a subsetPε′ of PCSε′ such that the canonical extensionj : PM

ε → Pε′26

is M-complete.27

We now define the almost finite support limit. (The direct limit will in general not do, as it may not28

contain the rangej[PMε ]. The almost finite support limit is the obvious modificationof the direct limit, and29

it is the smallest partial CS limitPε′ such thatj[PMε ] ⊆ Pε′ , and it indeed turns out to beM-complete as30

well.)31

Definition 4.13. Let ε be a limit ordinal inM, and letε′ ≔ sup(ε∩M). Let PM be a topped iteration inM32

of lengthε, and letP be a topless iteration inV of lengthε′. Assume that the canonical embeddingsiα33

work for all α ∈ ε ∩ M = ε′ ∩ M. Let iε be the canonical extension. We define thealmost finite support34

limit of P overPM (or: almost FS limit) as the following subforcingPε′ of PCSε′

:35

Pε′ ≔ q∧ iε(p) ∈ PCSε′ : p ∈ PM

ε and (∃α ∈ ε ∩ M) q ≤Pα iα(pα) .

Note that for cf(ε) > ω, the almost FS limit is equal to the direct limit, as eachp ∈ PMε is in fact inPM

α36

for someα ∈ ε ∩ M, soiε(p) = iα(p) ∈ Pα.37

Lemma 4.14. Assume thatP andPM are as above and let Pε′ be the almost FS limit. ThenPPε′ is a38

partial CS iteration, and iε works, i.e., iε is an M-complete embedding from PMε to Pε′ . (As Pε′ is a complete39

subforcing of Pε, this also implies that iε is M-complete from PMε to Pε.)1

32 For example: Letε = ω1 andε′ = ω1 ∩ M. Assume thatPMω1

is (in M) a (or: the unique) partial CS limit of a nontrivialiteration. Assume that we have a topless iterationP of lengthε′ in V such that the canonical embeddings work for allα ∈ ω1 ∩ M.If we setPε′ to be the full CS limit, then we cannot further extend it to anyiteration of lengthω1 such that the canonical embeddingiω1 works: Letpα andqα be as in footnote 31. InM, the setA = qα : α ∈ ω1 is a maximal antichain, and the sequence (pα)α∈ω1

is a decreasing coherent sequence. But inV there is an elementpε′ ∈ PCSε′

with pε′α = pα for all α ∈ ε ∩ M. This conditionpε′ is

clearly incompatible with all elements ofj[A] = j(pα) : α ∈ ε ∩ M. Hencej[A] is not maximal.

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Proof. It is easy to see thatPε′ is a partial CS limit and contains the rangeiε[PMε ]. We now show preserva-2

tion of predensity; this impliesM-completeness by Lemma 4.12.3

Let (p j) j∈J ∈ M be a maximal antichain inPMε , and letq∧ iε(p) be a condition inPε′ . (If ε′ < ε, i.e., if4

cf(ε) > ω, then we can choosep to be the empty condition.) Fixα ∈ ε∩M be such thatq ∈ Pα. Let Hα be5

Pα-generic and containq, so pα is in HMα . Now in M[HM

α ] the setp j : j ∈ J, p j ∈ PMε /H

Mα is predense6

in PMε /H

Mα (since this is forced by the empty condition inPM

α ). In particular,p is compatible with somep j ,7

witnessed byp′ ≤ p, p j in PMε /H

Mα .8

We can findq′ ≤Pα q deciding j andp′; since certainlyq′ ≤∗ iα(p′α), we may assume even≤ without9

loss of generality. Nowq′ ∧ iε(p′) ≤ q ∧ iε(p) (sinceq′ ≤ q andp′ ≤ p), andq′ ∧ iε(p′) ≤ iε(p j) (since10

p′ ≤ p j). 11

Definition and Claim 4.15. Let PM be a topped partial CS iteration inM of lengthε. We can construct by12

induction onβ ∈ ε + 1 analmost finite support iterationP overPM (or: almost FS iteration) as follows:13

(1) As induction hypothesis we assume that the canonical embeddingiα works for allα ∈ β ∩ M. (So14

the notationM[HMα ] makes sense.)15

(2) Let β = α + 1. If α ∈ M, then we can use anyQα provided that (it is forced that)QMα is an16

M[HMα ]-complete subforcing ofQα. (If α < M, then there is no restriction onQα.)17

(3) Letβ ∈ M and cf(β) = ω. ThenPβ is the almost FS limit of (Pα,Qα)α<β overPMβ

.18

(4) Let β ∈ M and cf(β) > ω. ThenPβ is again the almost FS limit of (Pα,Qα)α<β overPMβ

(which19

also happens to be the direct limit).20

(5) For limit ordinals not inM, Pβ is the direct limit.21

So the claim includes that the resultingP is a (topped) partial CS iteration of lengthε over PM (i.e.,22

the canonical embeddingsiα work for all α ∈ (ε + 1)∩ M), where we only assume that theQα satisfy the23

obvious requirement. (Note that we can always find some suitable Qα for α ∈ M, for example we can just24

takeQMα itself.)25

Proof. We have to show (by induction) that the resulting sequenceP is a partial CS iteration, and thatPM26

embeds intoP. For successor cases, there is nothing to do. So assume thatα is a limit. If Pα is a direct27

limit, it is trivially a partial CS limit; if Pα is an almost FS limit, then the easy part of Lemma 4.14 shows28

that it is a partial CS limit.29

So it remains to show that for a limitα ∈ M, the (naturally defined) embeddingiα : PMα → Pα is30

M-complete. This was the main claim in Lemma 4.14. 31

The following lemma is natural and easy.32

Lemma 4.16. Assume that we construct an almost FS iterationP overPM where each Qα is (forced to be)33

ccc. Then Pε is ccc (and in particular proper).34

Proof. We show thatPα is ccc by induction onα ≤ ε. For successors, we use thatQα is ccc. Forα of35

uncountable cofinality, we know that we took the direct limitcoboundedly often (and allPβ are ccc for36

β < α), so by a result of SolovayPα is again ccc. Forα a limit of countable cofinality not inM, just use37

that all Pβ are ccc forβ < α, and the fact thatPα is the direct limit. This leaves the case thatα ∈ M has38

countable cofinality, i.e., thePα is the almost FS limit. LetA ⊆ Pα be uncountable. Eacha ∈ A has the39

form q∧ iα(p) for p ∈ PMα andq ∈

⋃

γ<α Pγ. We can thin out the setA such thatp are the same and allq40

are in the samePγ. So there have to be compatible elements inA. 41

All almost FS iterations that we consider in this paper will satisfy the countable chain condition (and42

hence in particular be proper).43

We will need a variant of this lemma forσ-centered forcing notions.44

Lemma 4.17. Assume that we construct an almost FS iterationP overPM where only countably many Qα45

are nontrivial (e.g., only those withα ∈ M) and where each Qα is (forced to be)σ-centered. Then Pε is46

σ-centered as well.47

Proof. By induction: The direct limit of countably manyσ-centered forcings isσ-centered, as is the almost48

FS limit ofσ-centered forcings (to colorq∧ iα(p), usep itself together with the color ofq). 1

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We will actually need two variants of the almost FS construction: Countably many modelsMn; and2

starting the almost FS iteration with someα0.3

Firstly, we can construct an almost FS iteration not just over one iterationPM, but over an increasing4

chain of iterations. Analogously to Definition 4.13 and Lemma 4.14, we can show:5

Lemma 4.18. For each n∈ ω, let Mn be a nice candidate, and letPn be a topped partial CS iteration6

in Mn of length33 ε ∈ M0 of countable cofinality, such that Mm ∈ Mn and Mn thinks thatPm canonically7

embeds intoPn, for all m < n. LetP be a topless iteration of lengthε into which allPn canonically embed.8

Then we can define the almost FS limit Pε over (Pn)n∈ω as follows: Conditions in Pε are of the form9

q∧ inε(p) where n∈ ω, p ∈ Pnε, and q∈ Pα for someα ∈ Mn∩ ε. Then Pε is a partial CS limit over eachPn.10

As before, we get the following corollary:11

Corollary 4.19. Given Mn and Pn as above, we can construct a topped partial CS iterationP such that12

eachPn embeds Mn-completely into it; we can choose Qα as we wish (subject to the obvious restriction13

that each Qnα is an Mn[Hn

α]-complete subforcing). If we always choose Qα to be ccc, thenP is ccc; this is14

the case if we set Qα to be the union of the (countable) sets Qnα.15

Proof. We can definePα by induction. Ifα ∈⋃

n∈ω Mn has countable cofinality, then we use the almost16

FS limit as in Lemma 4.18. Otherwise we use the direct limit. If α ∈ Mn has uncountable cofinality,17

thenα′ ≔ sup(α ∩ M) is an element ofMn+1. In our induction we have already consideredα′ and have18

definedPα′ by Lemma 4.18 (applied to the sequence (Pn+1, Pn+2, . . .)). This is sufficient to show that19

inα : Pnα → Pα′ ⋖ Pα is Mn-complete. 20

Secondly, we can start the almost FS iteration after someα0 (i.e., P is already given up toα0, and we21

can continue it as an almost FS iteration up toε), and get the same properties that we previously showed22

for the almost FS iteration, but this time for the quotientPε/Pα0. In more detail:23

Lemma 4.20. Assume thatPM is in M a (topped) partial CS iteration of lengthε, and thatP is in V a24

topped partial CS iteration of lengthα0 over PMα0 for someα0 ∈ ε ∩ M. Then we can extendP to a25

(topped) partial CS iteration of lengthε over PM, as in the almost FS iteration (i.e., using the almost FS26

limit at limit pointsβ > α0 with β ∈ M of countable cofinality; and the direct limit everywhere else). We27

can use any Qα for α ≥ α0 (provided QMα is an M[HM

α ]-complete subforcing of Qα). If all Qα are ccc, then28

Pα0 forces that Pε/Hα0 is ccc (in particular proper); if moreover all Qα areσ-centered and only countably29

many are nontrivial, then Pα0 forces that Pε/Hα0 isσ-centered.30

4.D. Almost countable support iterations. “Almost countable support iterationsP” (over a given itera-31

tion PM in a candidateM) will have the following two crucial properties: There is a canonicalM-complete32

embedding ofPM into P, and P preserves a given random real (similar to the usual countable support33

iterations).34

Definition and Claim 4.21. Let PM be a topped partial CS iteration inM of lengthε. We can construct by35

induction onβ ∈ ε + 1 thealmost countable support iterationP overPM (or: almost CS iteration):36

(1) As induction hypothesis, we assume that the canonical embeddingiα works for everyα ∈ β ∩ M.37

We set3438

(4.22) δ ≔ min(M \ β), δ′ ≔ sup(α + 1 : α ∈ δ ∩ M).

Note thatδ′ ≤ β ≤ δ.39

(2) Letβ = α + 1. We can choose any desired forcingQα; if β ∈ M it is of course required thatQMα is40

anM[HMα ]-complete subforcing ofQα. This definesPβ.41

(3) Let cf(β) > ω. ThenPβ is the direct limit.42

(4) Let cf(β) = ω and assume thatβ ∈ M (soM ∩ β is cofinal inβ andδ′ = β = δ). We definePβ = Pδ43

as the union of the following two sets:44

• The almost FS limit of (Pα,Qα)α<δ, see Definition 4.13.1

33Or only: ε ∈ Mn0 for somen0.34 So for successorsβ ∈ M, we haveδ′ = β = δ. Forβ ∈ M limit, β = δ andδ′ is as in Definition 4.9.

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• The setPgenδ

of M-generic conditionsq ∈ PCSδ

, i.e., those which satisfy2

q PCSδ

i−1δ [HPCS

δ] ⊆ PM

δ is M-generic.

(5) Let cf(β) = ω and assume thatβ < M but M ∩ β is cofinal inβ, soδ′ = β < δ. We definePβ = Pδ′3

as the union of the following two sets:4

• The direct limit of (Pα,Qα)α<δ′ .5

• The setPgenδ′

of M-generic conditionsq ∈ PCSδ′

, i.e., those which satisfy6

q PCSδ′

i−1δ [HPCS

δ′] ⊆ PM

δ is M-generic.

(Note that theM-generic conditions form an open subset ofPCSβ= PCS

δ′.)

7(6) Let cf(β) = ω andM ∩ β not cofinal inβ (soβ < M). ThenPβ is the full CS limit of (Pα,Qα)α<β8

(see Definition 4.5).9

So the claim is that for every choice ofQα (with the one obvious restriction), this construction always10

results in a partial CS iterationP overPM. The proof is a bit cumbersome; it is a variant of the usual proof11

that properness is preserved in countable support iterations (see e.g. [Gol93]).12

We will use the following fact inM (for the iterationPM):13

(4.23)

Let P be a topped iteration of lengthε. Let α1 ≤ α2 ≤ β ≤ ε. Let p1 be aPα1-namefor a condition inPε, and letD be an open dense set ofPβ. Then there is aPα2-namep2 for a condition inD such that the empty condition ofPα2 forces: p2 ≤ p1β and:if p1 is in Pε/Hα2, then the conditionp2 is as well.

The following easy fact will also be useful:14

(4.24)Let P be a subforcing ofQ. We definePp ≔ r ∈ P : r ≤ p. Assume thatp ∈ PandPp = Qp.Then for anyP-name

˜x and any formulaϕ(x) we have:p P ϕ(

˜x) iff p Q ϕ(

˜x).

We now prove by induction onβ ≤ ε the following statement (which includes that the Definitionand15

Claim 4.21 works up toβ). Let δ, δ′ be as in (4.22).16

Lemma 4.25. (a) The topped iterationP of lengthβ is a partial CS iteration.17

(b) The canonical embedding iδ : PMδ→ Pδ′ works, hence also iδ : PM

δ→ Pδ works.

18

(c) Moreover, assume that19

• α ∈ M ∩ δ,20

•˜p ∈ M is a PM

α -name of a PMδ

-condition,21

• q ∈ Pα forces (in Pα) that˜pα[HM

α ] is in HMα .

22

Then there is a q+ ∈ Pδ′ (and therefore in Pβ) extending q and forcing that˜p[HM

α ] is in HMδ

.23

Proof. First let us deal with the trivial cases. It is clear that we always get a partial CS iteration.24

• Assume thatβ = β0 + 1 ∈ M, i.e.,δ = δ′ = β. It is clear thatiβ works. To getq+, first extendq to25

someq′ ∈ Pβ0 (by induction hypothesis), then defineq+ extendingq′ by q+(β0) ≔˜p(β0).26

• If β = β0 + 1 < M, there is nothing to do.27

• Assume that cf(β) > ω (whetherβ ∈ M or not). Thenδ′ < β. Soiδ : PMδ→ Pδ′ works by induction,28

and similarly (c) follows from the inductive assumption. (Use the inductive assumption forβ = δ′;29

theδ that we got at that stage is the same as the currentδ, and theq+ we obtained at that stage will30

still satisfy all requirements at the current stage.)31

• Assume that cf(β) = ω and thatM ∩ β is bounded inβ. Then the proof is the same as in the32

previous case.33

We are left with the cases corresponding to (4) and (5) of Definition 4.21: cf(β) = ω andM∩β is cofinal34

in β. So eitherβ ∈ M, thenδ′ = β = δ, or β < M, thenδ′ = β < δ and cf(δ) > ω.35

We leave it to the reader to check thatPβ is indeed a partial CS iteration. The main point is to see that36

q∧ p ∈ Pβ wheneverq ≤ pα. If p ∈ Pgenβ

, then this follows becausePgenβ

is open inPCSβ

; the other cases37

are immediate from the definition (by induction).38

We now turn to claim (c). Assumeq ∈ Pα and˜p ∈ M are given,α ∈ M ∩ δ.

39

Let (Dn)n∈ω enumerate all dense sets ofPMδ

which lie in M, and let (αn)n∈ω be a sequence of ordinals in1

M which is cofinal inβ, whereα0 = α.

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2

Using (4.23) inM, we can find a sequence (˜pn)n∈ω satisfying the following inM, for all n > 0:

3

•˜p0 =

˜p.

4

•˜pn ∈ M is aPM

αn-name of aPM

δ-condition inDn.

5• PM

αn ˜pn ≤PM

δ ˜pn−1.

6

• PMαn

If˜pn−1αn ∈ HM

αn, then

˜pnαn ∈ HM

αnas well.

7

Using the inductive assumption for theαn’s, we can now find a sequence (qn)n∈ω of conditions satisfying8

the following:9

• q0 = q, qn ∈ Pαn.10

• qnαn−1 = qn−1.11

• qn Pαn˜pn−1αn ∈ HM

αn, so also

˜pnαn ∈ HM

αn.

12

Let q+ ∈ PCSβ

be the union of theqn. Then for alln:13

(1) qn PCSβ ˜

pnαn ∈ HMαn

, so alsoq+ forces this.14

(Using induction onn.)15

(2) For alln and allm≥ n: q+ PCSβ ˜

pmαm ∈ HMαm

, so also˜pnαm ∈ HM

αm.16

(As˜pm ≤

˜pn.)

17

(3) q+ PCSβ ˜

pn ∈ HMδ

.18

(Recall thatPCSβ

is separative, see Fact 4.7. Soiδ(˜pn) ∈ Hδ iff iαn(

˜pαm) ∈ Hαm for all largem.)

19

As q+ PCSβ ˜

pn ∈ Dn ∩ HMδ

, we conclude thatq+ ∈ Pgenβ

(using 4.12, applied toPCSβ

). In particular,Pgenβ

20

is dense inPβ: Let q∧ iδ(p) be an element of the almost FS limit; soq ∈ Pα for someα < β. Now find a21

genericq+ extendingq and stronger thaniδ(p), thenq+ ≤ q∧ iδ(p).22

It remains to show thatiδ is M-complete. LetA ∈ M be a maximal antichain ofPMδ

, andp ∈ Pβ. Assume23

towards a contradiction thatp forces inPβ that i−1δ

[Hβ] does not intersectA in exactly one point.24

SincePgenβ

is dense inPβ, we can find someq ≤ p in Pgenβ

. Let25

P′ ≔ r ∈ PCSβ : r ≤ q = r ∈ Pβ : r ≤ q,

where the equality holds becausePgenβ

is open inPCSβ

.26

Let Γ be the canonical name for aP′-generic filter, i.e.:Γ ≔ (r, r) : r ∈ P′. Let R be eitherPCSβ

or Pβ.27

We write〈Γ〉R for the filter generated byΓ in R, i.e.,〈Γ〉R := r ∈ R : (∃r ′ ∈ Γ) r ′ ≤ r. So28

(4.26) q R HR = 〈Γ〉R.

We now see that the following hold:29

– q Pβ i−1δ

[HPβ ] does not intersectA in exactly one point. (By assumption.)30

– q Pβ i−1δ

[〈Γ〉Pβ ] does not intersectA in exactly one point. (By (4.26).)31

– q PCSβ

i−1δ

[〈Γ〉Pβ ] does not intersectA in exactly one point. (By (4.24).)32

– q PCSβ

i−1δ

[〈Γ〉PCSβ

] does not intersectA in exactly one point. (Becauseiδ mapsA into Pβ ⊆ PCSβ

, so33

A∩ i−1δ

[〈Y〉Pβ ] = A∩ i−1δ

[〈Y〉PCSβ

] for all Y.)34

– q PCSβ

i−1δ

[HPCSβ

] does not intersectA in exactly one point. (Again by (4.26).)35

But this, according to the definition ofPgenβ

, impliesq < Pgenβ

, a contradiction. 36

We can also show that the almost CS iteration of proper forcingsQα is proper. (We do not really need37

this fact, as we could allow non-proper iterations in our preparatory forcing, see Section 7.A(4). In some38

sense,M-completeness replaces properness, so the proof ofM-completeness was similar to the “usual”39

proof of properness.)40

Lemma 4.27. Assume that in Definition 4.21, every Qα is (forced to be) proper. Then also each Pδ is41

proper.42

Proof. By induction onδ ≤ ε we prove that for allα < δ the quotientPδ/Hα is (forced to be) proper. We43

use the following facts about properness:1

(4.28) If P is proper andP forces thatQ is proper, thenP ∗ Q is proper.

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2

(4.29)If P is an iteration of lengthω and if eachQn is forced to be proper, then the inverselimit Pω is proper, as are all quotientsPω/Hn.

3

(4.30)If P is an iteration of lengthδ with cf(δ) > ω, and if all quotientsPβ/Hα (for α <β < δ) are forced to be proper, then the direct limitPδ is proper, as are all quotientsPδ/Hα.

If δ is a successor, then our inductive claim easily follows fromthe inductive assumption together4

with (4.28).5

Let δ be a limit of countable cofinality, sayδ = supn δn. Define an iterationP′ of lengthω with6

Q′n ≔ Pδn+1/Hδn. (EachQ′n is proper, by inductive assumption.) There is a natural forcing equivalence7

betweenPCSδ

andP′CSω , the full CS limit of P′.

8

Let N ≺ H(χ∗) containP,Pδ, P′,M, PM. Let p ∈ Pδ ∩ N. Without loss of generalityp ∈ Pgenδ

. So below9

p we can identifyPδ with PCSδ

and hence withP′CSω ; now apply (4.29).

10

The case of uncountable cofinality is similar, using (4.30) instead. 11

Recall the definition of⊏n and⊏ and the notion of interpretationZ∗ (of a code˜Z for a null set) from12

Definition 2.25. We will need the following (“local”) variant of “random preservation”:13

Definition 4.31. Fix a modelM, a realr ∈ 2ω andQM ∈ M. Let QM be anM-complete subforcing ofQ.14

We say that “Q locally preserves randomness of r over M”, if there is a sequence (DQM

k )k∈ω in M of open15

dense subsets ofQM such that the following holds:16

Assume that17

• M thinks thatp≔ (pk)k∈ω interprets˜Z asZ∗ (so

˜Z is aQM-name of a code for a null set andZ∗ is18

a code for a null set, both inM),19

• moreover, eachpk is in DQM

k (we call such a sequence (pk)k∈ω, or the according interpretation,20

“quick”),21

• r is random overM (so there is a minimalc such thatZ∗ ⊏c r).22

Then there is aq ≤Q p0 forcing that23

• r is random overM[GM],24

•˜Z ⊏c r.25

Note that this is trivially satisfied ifr is not random overM.26

This property is a “local” variant of strong preservation ofrandomness (see Definition 2.27). For a27

variant of this definition, see Section 7.28

Lemma 4.32. • If QM is an ultralaver forcing in M and r a real, then there is an ultralaver forcing29

Q over35 QM locally preserving randomness of r over M.30

• If QM is a Janus forcing in M and r a real, then there is a Janus forcing Q over QM (which is in31

fact equivalent to random forcing) locally preserving randomness of r over M.32

Proof. Assume thatr is random overM (otherwise the claim is vacuous).33

Ultralaver: This follows directly from Lemma 2.29, settingDQM

n to be the set of conditions with stem34

of length at leastn.35

Janus:In this case, the notion of “quick” interpretations will be trivial, i.e., DQM

k = QM for all k.36

According to Lemma 2.43, we can first find some countableM′ such thatr is still random overM′ and37

such thatM′ thinks that 2ω∩M is countable. InM′ the setQM is now a countable Janus forcing; so we can38

apply Lemma 3.16 to construct a Janus forcingQM′ overQM which is equivalent to random forcing. InV,39

let Q be random forcing; since this is a Suslin ccc forcing we know thatQM′ is anM′-complete subforcing40

of Q. Moreover, as was noted in Lemma 2.28, we know thatQ strongly preserves randoms overM′.41

So assume that (inM) the sequence (pk)k∈ω of QM-conditions interprets˜Z asZ∗. Since (inM′), QM

42

is M-complete inQM′ , also inM′ the QM′ -name˜Z is interpreted asZ∗. Let c be such thatZ∗ ⊏c r. So43

by strong preservation of randoms, we can inV find someq ≤ p0 forcing thatr is random overM′[HM′ ]1

35“Q overQM” just means thatQM is anM-complete subforcing ofQ.

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(and therefore also over the subsetM[HM]), and that˜Z ⊏c r (where

˜Z can be evaluated inM′[HM′ ] or2

equivalently inM[HM]). 3

We will prove the following preservation theorem:4

Lemma 4.33. Let P be an almost CS iteration (of lengthε) over PM, r random over M, and p∈ PMε .5

Assume that each Pα forces that Qα locally preserves randomness of r over M[HMα ]. Then there is some6

q ≤ p in Pε forcing that r is random over M[HMε ].7

What we will actually need is the following variant:8

Lemma 4.34. Assume thatPM is in M a topped almost CS iteration of lengthε, and we already have some9

topped partial CS iterationP overPMα0 of lengthα0 ∈ M ∩ ε. Let˜r be a Pα0-name of a random real10

over M[HMα0

]. Assume that we extendP to lengthε as an almost CS iteration36 using forcings Qα which11

locally preserve the randomness of˜r over M, witnessed by a sequence(DQM

α

k )k∈ω. Let p∈ PMε . Then we can12

find a q≤ p in Pε forcing that˜r is random over M[HM

ε ].13

Actually, we will only prove the two previous lemmas under the following additional assumption (which14

is enough for our application, and saves some unpleasant work). This additional assumption is not really15

necessary; without it, we could use the method of [GK06] for the proof.16

Assumption 4.35. • For eachα ∈ M ∩ ε, (PMα forces that)QM

α is either trivial37 or adds a newω-17

sequence of ordinals. Note that in the latter case we can assume without loss of generality that18⋂

n∈ω DQM

n = ∅ (and, of course, that theDQM

n are decreasing).19

• Moreover, we assume that already inM there is a setT ⊆ ε such thatPα forces:Qα is trivial iff20

α ∈ T. (So whetherQα is trivial or not does not depend on the generic filter belowα, it is already21

decided in the ground model.)22

The result will follow as a special case of the following lemma, which we prove by induction onβ.23

(Note that this is a refined version of the proof of Lemma 4.25 and similar to the proof of the preservation24

theorem in [Gol93, 5.13].)25

Definition 4.36. Under the assumptions of Lemma 4.34 and Assumption 4.35, let˜Z be aPδ-name,α0 ≤26

α < δ, and letp = (pk)k∈ω be a sequence ofPα-names of conditions inPδ/Hα. Let Z∗ be aPα-name.27

We say that ( ¯p,Z∗) is a quick interpretation of˜Z if p interprets

˜Z asZ∗ (i.e., Pα forces thatpk forces28

˜Zk = Z∗k for all k), and moreover:29

Lettingβ ≥ α be minimal withQMβ

nontrivial (if suchβ exists):Pβ forces that the sequence30

(pk(β))k∈ω is quick inQMβ

, i.e., pk(β) ∈ DQMβ

k for all k.31

It is easy to see that:32

(4.37) For every name˜Z there is a quick interpretation ( ¯p,Z∗).

Lemma 4.38. Under the same assumptions as above, letβ, δ, δ′ be as in(4.22)(so in particular we have33

δ′ ≤ β ≤ δ ≤ ε).34

Assume that35

• α ∈ M ∩ δ (= M ∩ β) andα ≥ α0 (soα < δ′),36

• p ∈ M is a PMα -name of a PM

δ-condition,

37

•˜Z ∈ M is a PM

δ-name of a code for null set,38

• Z∗ ∈ M is a PMα -name of a code for a null set,39

• PMα forces: p = (pk)k∈ω ∈ M is a quick sequence in PM

δ/HMα interpreting

˜Z as Z∗ (as in Defini-40

tion 4.36),41

• PMα forces: if pα ∈ HM

α , then p0 ≤ p,42

• q ∈ Pα forces pα ∈ HMα ,1

36Of course our official definition of almost CS iteration assumes that we start the construction at 0, so we modify this definitionin the obvious way.

37More specifically,QMα = ∅.

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• q forces that r is random over M[HMα ], so in particular there is a Pα-name c0 in V for the minimal c2

with Z∗ ⊏c r.3

Then there is a condition q+ ∈ Pδ′ , extending q, and forcing the following:4

• p ∈ HMδ

,5

• r is random over M[HMδ

],6

•˜Z ⊏

˜c r.

7

We actually claim a slightly stronger version, where instead of Z∗ and˜Z we have finitely many codes8

for null sets and names of codes for null sets, respectively.We will use this stronger claim as inductive9

assumption, but for notational simplicity we only prove theweaker version; it is easy to see that the weaker10

version implies the stronger version.11

Proof. “Nontrivial” successor case:β = γ + 1 ∈ M.12

If QMγ is trivial, there is nothing to do.

13

Assume thatγ ≥ α is minimal such thatQMγ is a nontrivial forcing notion. Work inV[Hγ] whereq ∈ Hγ.14

Note thatM[HMγ ] = M[HM

α ]. Sor is random overM[HMγ ], and (pk(γ))k∈ω quickly interprets

˜Z asZ∗ in QM

γ .15

Now let q+γ = q, and use the fact thatQγ locally preserves randomness to findq+(γ) ≤ p0(γ).16

Assume thatQMγ is nontrivial, but thatγ is not minimal. Again work inV[Hγ]. Let k∗ be maximal with17

pk∗γ ∈ HMγ . Consider

˜Z as aQM

γ -name, and (using (4.37)) find a quick interpretationZ′ of˜Z witnessed by18

a sequence starting withpk∗ (γ). In M[HMα ], Z′ is now aPM

γ /HMα -name. Clearly, the sequence (pkγ)k∈ω is19

a quick sequence interpretingZ′ asZ∗. (Use the fact thatpkγ forcesk∗ ≥ k.)20

Using the induction hypothesis, we can first extendq to a conditionq′ ∈ Pγ and then (again by our21

assumption thatQγ locally preserves randomness) to a conditionq+ ∈ Pγ+1.22

The nontrivial limit case: M ∩ β unbounded inβ, i.e., δ′ = β. (This deals with cases (4) and (5) in23

Definition 4.21. In case (4) we haveβ ∈ M, i.e.,β = δ; in case (5) we haveβ < M andβ < δ.)24

Let α = δ0 < δ1 < · · · be a sequence ofM-ordinals cofinal inM ∩ δ′ = M ∩ δ. We may assume38 that25

eachQMδn

is nontrivial.26

Let (˜Zn)n∈ω be a list of allPM

δ-names inM for codes for null sets (starting with our given null set

˜Z =

˜Z0).27

Let (En)n∈ω enumerate all open dense sets ofPMδ

from M, without loss of generality39 we can assume that:28

(4.39) En decides˜Z0n, . . . ,

˜Znn.

We write pk0 for pk, andZ0,0 for Z∗; as mentioned above,

˜Z =

˜Z0.

29

By induction onn we can now find a sequence ¯pn = (pkn)k∈ω and PM

δn-namesZi,n for i ∈ 0, . . . , n30

satisfying the following:31

(1) PMδn

forces thatp0n ≤ pk

n−1 wheneverpkn−1 ∈ PM

δ/HMδn

.32

(2) Pαn forces thatp0n ∈ En. (ClearlyEn ∩ PM

δ/HMαn

is a dense set.)33

(3) pn ∈ M is aPMδn

-name for a quick sequence interpreting (˜Z0, . . . ,

˜Zn) as (Z0,n, . . . ,Zn,n) (in PM

δ/HMδn

),34

soZi,n is aPMδn

-name of a code for a null set, for 0≤ i ≤ n.35

Note that this implies that the sequence (pkn−1δn) is (forced to be) a quick sequence interpreting (Z0,n, . . . ,Zn−1,n)36

as (Z0,n−1, . . . ,Zn−1,n−1) .37

Using the induction hypothesis, we now define a sequence (qn)n∈ω of conditionsqn ∈ Pδn and a sequence38

(cn)n∈ω (wherecn is aPδn-name) such that (forn > 0) qn extendsqn−1 and forces the following:39

• p0n−1δn ∈ HM

δn.

40

• Therefore,p0n ≤ p0

n−1.41

• r is random overM[HMδn

].42

• Let cn be the leastc such thatZn,n ⊏c r.43

• Zi,n ⊏ci r for i = 0, . . . , n− 1.44

Now let q =⋃

n qn ∈ PCSδ′

. As in Lemma 4.25 it is easy to see thatq ∈ Pgenδ′⊆ Pδ′ . Moreover, by (4.39) we45

get thatq forces that˜Zi = limn Zi,n. Since each setCc,r := x : x ⊏c r is closed, this implies thatq forces1

˜Zi ⊏ci r, in particular

˜Z =

˜Z0 ⊏c0 r.

38If from someγ on all QMζ

are trivial, thenPMδ= PM

γ , so by induction there is nothing to do.39well, if we just enumerate a basis of the open sets instead of all of them. . .

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2

The trivial cases:In all other cases,M ∩ β is bounded inβ, so we already dealt with everything at stage3

β0 ≔ sup(β ∩ M). Note thatδ′0 andδ0 used at stageβ0 are the same as the currentδ′ andδ. 4

5. T 5

In this section we describe aσ-closed “preparatory” forcing notionR; the generic filter will define a6

“generic” forcing iterationP, so elements ofR will be approximations to such an iteration. In Section 6 we7

will show that the forcingR ∗ Pω2 forces BC and dBC.8

From now on, we assume CH in the ground model.9

5.A. Alternating iterations, canonical embeddings and the preparatory forcing R. The preparatory10

forcingR will consist of pairs (M, P), whereM is a countable model andP ∈ M is an iteration of ultralaver11

and Janus forcings.12

Definition 5.1. An alternating iteration40 is a topped partial CS iterationP of lengthω2 satisfying the13

following:14

• EachPα is proper.4115

• For α even, either bothQα and Qα+1 are (forced by the empty condition to be) trivial,42 or Pα16

forces thatQα is an ultralaver forcing adding the generic realℓα, andPα+1 forces thatQα+1 is a17

Janus forcing based onℓ∗α (whereℓ∗ is defined fromℓ as in Lemma 2.22).18

We will call an even index an “ultralaver position” and an oddone a “Janus position”.19

As in any partial CS iteration, eachPδ for cf(δ) > ω (and in particularPω2) is a direct limit.20

Recall that in Definition 4.10 we have defined the notion “PM canonically embeds intoP” for nice21

candidatesM and iterationsP ∈ V andPM ∈ M. Since our iterations now have lengthω2, this means that22

the canonical embedding works up to and including43ω2.23

In the following, we will use pairsx = (Mx, Px) as conditions in a forcing, wherePx is an alternating24

iteration in the nice candidateMx. We will adapt our notation accordingly: Instead of writingM, PM, PMα25

HMα (the induced filter),QM

α , etc, we will writeMx, Px, Pxα, Hx

α, Qxα, etc. Instead of “Px canonically embeds26

into P” we will say “x canonically embeds intoP” (which is a more exact notation anyway, since the test27

whether the embedding isMx-complete uses bothMx andPx, not justPx).28

The following rephrases Definition 4.10 of a canonical embedding in our new notation, taking into29

account that:30

LDx is anMx-complete subforcing ofLD iff D extendsDx31

(see (5) of the ultralaver wish list 2.1).32

Fact 5.2. x = (Mx, Px) canonically embeds intoP, if (inductively) for allβ ∈ ω2∩Mx∪ω2 the following33

holds:34

• Let β = α + 1 for alpha even (i.e., an ultralaver position). Then eitherQxα is trivial (andQα can35

be trivial or not), or we require that (Pα forces that) theV[Hα]-ultrafilter systemD used forQα36

extends theMx[Hxα]-ultrafilter systemDx used forQx

α.37

• Let β = α+ 1 for alpha odd (i.e., a Janus position). Then eitherQxα is trivial, or we require that (Pα38

forces that) the Janus forcingQxα is anMx[Hx

α]-complete subforcing of the Janus forcingQα.39

• Let β be a limit. Then the canonical extensioniβ : Pxβ→ Pβ is Mx-complete. (The canonical40

extension was defined in Definition 4.9.)41

Fix a sufficiently large regular cardinalχ∗.1

40See Section 7 for possible variants of this definition.41This does not seem to be necessary, see Section 7, but it is easy to ensure and might be comforting to some of the readers and/or

authors.42For definiteness, let us agree that the trivial forcing is thesingleton∅.43This is stronger than to require that the canonical embedding works for everyα ∈ ω2 ∩ M, even though bothPω2 andPM

ω2are

just direct limits; see footnote 32.

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Definition 5.3. The “preparatory forcing” R consists of pairsx = (Mx, Px) such thatMx ∈ H(χ∗) is a nice2

candidate containingω2, andPx is in Mx an alternating iteration (in particular topped and of lengthω2).3

We definey to be stronger thanx (in symbols:y ≤R x), if the following holds: eitherx = y, or:4

• Mx ∈ My andMx is countable inMy.5

• My thinks thatx canonically embeds intoPy.6

We will sometimes writeix,y for the canonical embedding (inMy) from Pxω2

to Pyω2

.7

There are several variants of this definition which result inequivalent forcing notions. We will briefly8

come back to this in Section 7.9

Note that the order onR is transitive.10

The following is trivial by elementarity:11

Fact 5.4. Assume thatP is an alternating iteration (inV), thatx ∈ R canonically embeds intoP, and that12

N ≺ H(χ∗) containsx andP. Let y = (My, Py) be the ord-collapse of (N, P). Theny ∈ R andy ≤ x.13

This fact will be used, for example, to get from the followingLemma 5.5 to Corollary 5.6.14

Lemma 5.5. Given x∈ R, there is an alternating iterationP such that x canonically embeds intoP.15

Proof. For the proof, we use either of the partial CS constructions introduced in the previous chapter (i.e.,16

an almost CS iteration or an almost FS iteration overPx). The only thing we have to check is that we can17

indeed chooseQα that satisfy the definition of an alternating iteration (i.e., as ultralaver or Janus forcings)18

and such thatQxα is Mx-complete inQα.19

In the ultralaver case we arbitrarily extendDx to an ultrafilter systemD, which is justified by ultralaver20

wish list 2.1 item (5).21

In the Janus case, we takeQα ≔ Qxα (this works by Janus wish list 3.1 item (3). Alternatively, we could22

extendQxα to a random forcing, by Janus wish list item (4). 23

Corollary 5.6. Given x∈ R and an HCON object b∈ H(χ∗) (e.g., a real or an ordinal), there is a y≤ x24

such that b∈ My.25

What we will actually need are the following three variants:26

Lemma 5.7. (1) Given x∈ R there is aσ-centered alternating iterationP above x.27

(2) Given a decreasing sequencex = (xn)n∈ω in R, there is an alternating iterationP such that each28

xn embeds intoP. Moreover, we can assume that for all Janus positionsβ, the Janus44 forcing Qβ29

is (forced to be) the union of the Qxnβ

, and that for all limitsα, the forcing Pα is the almost FS limit30

over(xn)n∈ω (as in Corollary 4.19).31

(3) Let x, y ∈ R. Let jx be the transitive collapse of Mx, and define jy analogously. Assume that32

jx[Mx] = jy[My], that jx(Px) = jy(Py) and that there areα0 ≤ α1 < ω2 such that:33

• Mx ∩ α0 = My ∩ α0 (and thus jxα0 = jyα0).34

• Mx ∩ [α0, ω2) ⊆ [α0, α1).35

• My ∩ [α0, ω2) ⊆ [α1, ω2).36

Then there is an alternating iterationP such that both x and y canonically embed into it.37

Proof. For (1), use an almost FS iteration. We only use the coordinates inM, use the (countable!) Janus38

forcingsQα ≔ Qxα for all Janus positionsα ∈ M. Ultralaver forcing areσ-centered anyway, soPε will be39

σ-centered, by Lemma 4.17.40

For (2), use the almost FS iteration over the sequence (xn)n∈ω as in Corollary 4.19, and at Janus positions41

α setQα to be the union of theQxnα . (By Fact 3.8,Qxn

α is Mxn-complete inQα, so Corollary 4.19 can be42

applied here.)43

For (3), we again use an almost FS construction. This time we start with an almost FS construction over44

x up toα1, and then continue with an almost FS construction overy. 45

As above, Fact 5.4 gives us the following consequences:46

Corollary 5.8. (1) R is σ-closed. HenceR does not add new HCON objects (and in particular: no47

new reals).1

44If all Qxnβ

are trivial, then we may also setQβ to be the trivial forcing, which is formally not a Janus forcing.

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(2) R forces that the generic filter G⊆ R isσ-directed, i.e., for every countable subset B of G there is2

a y ∈ G stronger than each element of B.3

(3) R forces CH. (Since we assume CH in V.)4

(4) Given a decreasing sequencex = (xn)n∈ω in R and any HCON object b∈ H(χ∗), there is a y∈ R5

such that6

• y ≤ xn for all n,7

• My contains b and the sequencex,8

• for all Janus positionsβ, My thinks that the Janus forcing Qyβ

is (forced to be) the union of9

the Qxnβ

,10

• for all limits α, My thinks that Pyα is the almost FS limit over(xn)n∈ω (of (Pyβ)β<α).11

Proof. Item (4) directly follows from Lemma 5.7(2) and Fact 5.4. Item (1) is a special case of (4), and (2)12

and (3) are trivial consequences of (1). 13

Another consequence of Lemma 5.7 is:14

Lemma 5.9. The forcing notionR is ℵ2-cc.15

Proof. Recall that we assume thatV (and henceV[G]) satisfies CH.16

Assume towards a contradiction that (xi : i < ω2) is an antichain. Using CH we may without loss of17

generality assume that for eachi ∈ ω2 the transitive collapse of (Mxi , Pxi ) is the same. SetLi ≔ Mxi ∩ ω2.18

Using the∆-lemma we find some uncountableI ⊆ ω2 such that theLi for i ∈ I form a∆-system with rootL.19

Setα0 = sup(L) + 3. Moreover, we may assume sup(Li) < min(L j \ α0) for all i < j.20

Now take anyi, j ∈ I , set x ≔ xi andy ≔ x j , and use Lemma 5.7(3). Finally, use Fact 5.4 to find21

z≤ xi , x j. 22

5.B. The generic forcing P′. Let G be R-generic. ObviouslyG is a ≤R-directed system. Using the23

canonical embeddings, we can construct inV[G] a direct limit P′ω2of the directed systemG: Formally,24

we set25

P′ω2≔ (x, p) : x ∈ G andp ∈ Px

ω2,

and we set (y, q) ≤ (x, p) if y ≤R x andq is (in y) stronger thanix,y(p) (whereix,y : Pxω2→ Py

ω2is the26

canonical embedding). Similarly, we define for eachα27

P′α := (x, p) : x ∈ G, α ∈ Mx andp ∈ Pxα

with the same order.28

To summarize:29

Definition 5.10. Forα ≤ ω2, the direct limit of thePxα with x ∈ G is calledP′α.30

Formally, elements ofP′ω2are defined as pairs (x, p). However, thex does not really contribute any31

information. In particular:32

Fact 5.11. (1) Assume that (x, px) and (y, py) are inP′ω2, thaty ≤ x, and that the canonical embedding33

ix,y witnessingy ≤ x mapspx to py. Then (x, px) =∗ (y, py).34

(2) (y, q) is in P′ω2stronger than (x, p) iff for some (or equivalently: for any)z ≤ x, y in G the canoni-35

cally embeddedq is in Pzω2

stronger than the canonically embeddedp. The same holds if “stronger36

than” is replaced by “compatible with” or by “incompatible with”.37

(3) If (x, p) ∈ P′α, and ify is such thatMy = Mx andPyα = Pxα, then (y, p) =∗ (x, p).38

In the following, we will therefore often abuse notation andjust write p instead of (x, p) for an element39

of P′α.40

We can define a natural restriction map fromP′ω2to P′α, by mapping (x, p) to (x, pα). Note that by the41

fact above, we can assume without loss of generality thatα ∈ Mx. More exactly: There is ay ≤ x in G42

such thatα ∈ My (according to Corollary 5.6). Then inP′ω2we have (x, p) =∗ (y, p).

43

Fact 5.12. The following is forced byR:44

• P′β

is completely embedded intoP′α for β < α ≤ ω2 (witnessed by the natural restriction map).45

• If x ∈ G, thenPxα is Mx-completely embedded intoP′α for α ≤ ω2 (by the identity mapp 7→ (x, p)).1

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• If cf(α) > ω, thenP′α is the union of theP′β

for β < α.2

• By definition,P′ω2is a subset ofV.

3

G will always denote anR-generic filter, while theP′ω2-generic filter overV[G] will be denoted byH′ω2

4

(and the inducedP′α-generic byH′α). Recall that for eachx ∈ G, the mapp 7→ (x, p) is anMx-complete5

embedding ofPxω2

into P′ω2(and ofPx

α into P′α). This wayH′α ⊆ P′α induces anMx-generic filterHxα ⊆ Px

α.6

Sox ∈ R forces thatP′α is approximated byPxα. In particular we get:7

Lemma 5.13.Assume that x∈ R, α ≤ ω2 in Mx, p ∈ Pxα, andτ is a Px

α-name in Mx. Then Mx |= p Pxαϕ(τ)8

iff x R (x, p) P′α Mx[Hxα] |= ϕ(τ[H

xα]).9

Proof. “⇒” is clear. So assume thatϕ(τ) is not forced inMx. Then someq ≤Pxα

p forces the negation. Now10

x forces that (x, q) ≤ (x, p) in P′α; but the conditions (x, p) and (x, q) force contradictory statements. 11

5.C. The inductive proof of ccc. We will now prove by induction onα thatP′α is (forced to be) ccc and12

(equivalent to) an alternating iteration. Once we know this, we can prove Lemma 5.25, which easily implies13

all the lemmas in this section. So in particular these lemmaswill only be needed to prove ccc and not for14

anything else (and they will probably not aid the understanding of the construction).15

In this section, we try to stick to the following notation:R-names are denoted with a tilde underneath16

(e.g.,˜τ), while Px

α-names orP′α-names (for anyα ≤ ω2) are denoted with a dot accent (e.g., ˙τ). We use17

both accents when we deal withR-names forP′α-names (e.g.,˜τ).18

We first prove a few lemmas that are easy generalizations of the following straightforward observation:19

Assume thatx R (˜z,

˜p) ∈ P′α. In particular,x

˜z ∈ G. We first strengthenx to somex1 that decides

˜z20

and˜p to bez∗ andp∗. Thenx1 ≤

∗ z∗, so we can further strengthenx1 to somey ≤ z∗. By definition, this21

means thatz∗ is canonically embedded intoPy; so (by Fact 5.11) thePz∗α -conditionp∗ can be interpreted as a22

Pyα-condition as well. So we end up with somey ≤ x and aPy

α-conditionp∗ such thaty R (˜z,

˜p) =∗ (y, p∗).

23SinceR isσ-closed, we can immediately generalize this to countably many (R-names for)P′α-conditions:24

Fact 5.14. Assume thatx R˜pn ∈ P′α for all n ∈ ω. Then there is ay ≤ x and there arep∗n ∈ Py

α such that25

y R˜pn =

∗ p∗n for all n ∈ ω.26

Recall that more formally we should write:x R (˜zn,

˜pn) ∈ P′α; andy R (

˜zn,

˜pn) =∗ (y, p∗n).27

We will need a variant of the previous fact:28

Lemma 5.15. Assume thatP′β

is forced to be ccc, and assume that x forces (inR) that˜rn is a P′

β-name for29

a real (or an HCON object) for every n∈ ω. Then there is a y≤ x and there are Pyβ-namesr∗n in My such30

that y R ( P′β ˜

rn = r∗n) for all n.31

(Of course, we mean:˜rn is evaluated byG ∗ H′

β, while r∗n is evaluated byHy

β.)

32

Proof. The proof is an obvious consequence of the previous fact, since names of reals in a ccc forcing can33

be viewed as a countable sequence of conditions.34

In more detail: For notational simplicity assume all˜rn are names for elements of 2ω. Working inV, we35

can find for eachn,m∈ ω names for a maximal antichainÃn,m and for a function

˜fn,m :

Ãn,m→ 2 such that36

x forces that (P′β

forces that)˜rn(m) =

˜fn,m(a) for the uniquea ∈

Ãn,m ∩ H′

β. SinceP′

βis ccc, each

Ãn,m is37

countable, and sinceR is σ-closed, it is forced that the sequence˜Ξ = (

Ãn,m,

˜fn,m)n,m∈ω is in V.

38In V, we strengthenx to x1 to decide

˜Ξ to be someΞ∗. We can also assume thatΞ∗ ∈ Mx1 (see39

Corollary 5.6). EachA∗n,m consists of countably manya such thatx1 forcesa ∈ P′β. Using Fact 5.1440

iteratively (and again the fact thatR is σ-closed) we get somey ≤ x1 such that each sucha is actually an41

element ofPyβ. So inMy, we can use (A∗n,m, f

∗n,m)n,m∈ω to constructPy

β-names ˙r∗n in the obvious way.

42

Now assume thaty ∈ G and thatH′β

is P′β-generic overV[G]. Fix anya ∈ A∗n,m = ˜

An,m. Sincea ∈ Pyβ, we43

geta ∈ Hyβ

iff a ∈ H′β. So there is a unique elementa of A∗n,m∩Hy

β, and ˙r∗n(m) = f ∗n,m(a) =

˜fn,m(a) =

˜rn(m).

44

We will also need the following modification:45

Lemma 5.16. (Same assumptions as in the previous lemma.) In V[G][H′β], setQβ be the union of Qz

β[Hzβ]46

for all z ∈ G. In V, assume that x forces that each˜rn is a name for an element ofQβ. Then there is a y≤ x47

and there is in My a sequence(r∗n)n∈ω of Pyβ-names for elements of Qy

βsuch that y forces

˜rn = r∗n for all n.

1

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So the difference to the previous lemma is: We additionally assume that˜rn is in

⋃

z∈G Qzβ, and we2

additionally get that ˙r∗n is a name for an element ofQyβ.

3

Proof. Assumex ∈ G and work inV[G]. Fix n. P′β

forces that there is someyn ∈ G and somePyn

β-name4

τn ∈ Myn of an element ofQyn

βsuch that

˜rn (evaluated byH′

β) is the same asτn (evaluated byHyn

β). Since5

we assume thatP′β

is ccc, we can find a countable setYn ⊆ G of the possibleyn, i.e., the empty condition6

of P′β

forcesyn ∈ Yn. (AsR isσ-closed andYn ⊆ R ⊆ V, we must haveYn ∈ V.)7

So in V, there is (for eachn) anR-name˜Yn for this countable set. SinceR is σ-closed, we can find8

somez0 ≤ x deciding each˜Yn to be some countable setY∗n ⊆ R. In particular, for eachy ∈ Y∗n we know9

thatz0 R y ∈ G, i.e.,z0 ≤∗ y; so using once again thatR is σ-closed we can find somez stronger thanz010

and all they ∈⋃

n∈ω Y∗n. Let X contain allτ ∈ My such that for somey ∈⋃

n∈ω Y∗n, τ is a Pyβ-name for a11

Qyβ-element. Sincez≤ y, eachτ ∈ X is actually45 a Pz

β-name for an element ofQz

β.

12SoX is a set ofPz

β-names forQz

β-elements; we can assume thatX ∈ Mz. Also, z forces that

˜rn ∈ X for13

all n. Using Lemma 5.15, we can additionally assume that there arenamesPzβ-name ˙r∗n in Mz such thatz14

forces that˜rn = r∗n is forced for eachn. By Lemma 5.13, we know thatMz thinks thatPz

βforces that ˙r∗n ∈ X.15

Therefore ˙r∗n is aPzβ-name for aQz

β-element.

16

We now prove by induction onα thatP′α is equivalent to a ccc alternating iteration:17

Lemma 5.17. The following holds in V[G] for α < ω2:18

(1) P′α is equivalent to an alternating iteration. More formally: There is an iteration(Pβ,Qβ)β<α19

with limit Pα that satisfies the definition of alternating iteration (up toα), and there is a naturally20

defined dense embedding jα : P′α → Pα, such that forβ < α we have jβ ⊆ jα, and the embeddings21

commute with the restrictions.46 EachQα is the union of all Qxα with x ∈ G. For x∈ G withα ∈ Mx,22

the function ix,α : Pxα → Pα that maps p to jα(x, p) is the canonical Mx-complete embedding.23

(2) In particular, aP′α-generic filter H′α can be translated into aPα-generic filter which we call Hα24

(and vice versa).25

(3) Pα has a dense subset of sizeℵ1.26

(4) Pα is ccc.27

(5) Pα forces CH.28

Proof. α = 0 is trivial (sinceP0 andP′0 both are trivial: P0 is a singleton, andP′0 consists of pairwise29

compatible elements).30

So assume that all items hold for allβ < α.31

Proof of (1).32

Ultralaver successor case:Letα = β+1 with β an ultralaver position. LetHβ bePβ-generic overV[G].33

Work in V[G][Hβ]. By induction, for everyx ∈ G the canonical embeddingjβ defines aPxβ-generic filter34

overMx calledHxβ.

35Definition ofQβ (and thus ofPα): In Mx[Hx

β], the forcing notionQx

βis defined asLDx for some system36

of ultrafiltersDx in Mx[Hxβ]. Fix somes ∈ ω<ω. If y ≤ x in G, thenDy

s extendsDxs. Let Ds be the union37

of all Dxs with x ∈ G. SoDs is a proper filter. It is even an ultrafilter: Letr be aPβ-name for a real. Using38

Lemma 5.15, we know that there is somey ∈ G and somePyβ-name

˜ry ∈ My such that (inV[G][Hβ]) we39

have˜ry[Hy

β] = r. Sor ∈ My[Hy

β], hence eitherr or its complement is inDy

s and therefore inDs. So all filters40

in the familyD = (Ds)s∈ω<ω are ultrafilters.41

Now work again inV[G]. We setQβ to be thePβ-name forLD. (Note thatPβ forces thatQβ literally is42

the union of theQxβ[Hxβ] for x ∈ G, again by Lemma 5.15.)

43Definition of jα: Let (x, p) be inP′α. If p ∈ Px

β, then we setjα(x, p) = jβ(x, p), i.e., jα will extend jβ. If44

p = (pβ, p(β)) is in Pxα but not inPx

β, we setjα(x, p) = (r, s) ∈ Pβ ∗ Qβ wherer = jβ(x, pβ) ands is the45

(Pα-name for)p(β) as evaluated inMx[Hxβ]. FromQβ =

⋃

x∈G Qxβ[Hxβ] we conclude that this embedding is46

dense.1

45Here we use two consequences ofz ≤ y: Every Pyβ-name inMy can be canonically interpreted as aPz

β-name inMz, andQy

βis

(forced to be) a subset ofQzβ.

46I.e., jβ(x, pβ) = jα(x, pβ) = jα(x, p)β.

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The canonical embedding:By induction we know thatix,β which mapsp ∈ Pxβ

to jβ(x, p) is (the restric-2

tion to Pxβ

of) the canonical embedding ofx into Pω2. So we have to extend the canonical embedding to3

ix,α : Pxα → Pα. By definition of “canonical embedding”,ix,α mapsp ∈ Px

α to the pair (ix,β(pβ), p(β)). This4

is the same asjα(x, p). We already know thatDxs is (forced to be) anMx[Hx

β]-ultrafilter that is extended5

by Ds.6

Janus successor case:This is similar, but simpler than the previous case: Here,Qβ is just defined as7

the union of allQxβ[Hxβ] for x ∈ G. We will show below that this union satisfies the ccc; just as in Fact 3.8,8

it is then easy to see that this union is again a Janus forcing.9

In particular,Qβ consists of hereditarily countable objects (since it is theunion of Janus forcings, which10

by definition consist of hereditarily countable objects). So sincePβ forces CH,Qβ is forced to have sizeℵ1.11

Also note that since all Janus forcings involved are separative, the union (which is a limit of an incom-12

patibility-preserving directed system) is trivially separative as well.13

Limit case:Let α be a limit ordinal.14

Definition ofPα and jα: First we definejα : P′α → PCSα : For each (x, p) ∈ P′α, let jα(x, p) ∈ Pβ be15

the union of all jβ(x, pβ) (for β ∈ α ∩ Mx). (Note thatβ1 < β2 implies that jβ1(x, pβ1) is restriction of16

jβ2(x, pβ2), so this union is indeed an element ofPCSα .)17

Pα is the set of allq∧ p, wherep ∈ jα[P′α], q ∈ Pβ for someβ < α, andq ≤ pβ.18

It is easy to check thatPα actually is a partial countable support limit, and thatjα is dense. We will19

show below thatPα satisfies the ccc, so in particular it is proper.20

The canonical embedding:To see thatix,α is the (restriction of the) canonical embedding, we just have21

to check thatix,α is Mx-complete. This is the case sinceP′α is the direct limit of allPyα for y ∈ G (without22

loss of generalityy ≤ x), and eachix,y is Mx-complete (see Fact 5.12).23

Proof of (3).24

Recall that we assume CH in the ground model.25

The successor case,α = β + 1, follows easily from (3)–(5) forPβ (sincePβ forces thatQβ has size26

2ℵ0 = ℵ1 = ℵV1 ).

27

If cf(α) > ω, thenPα =⋃

β<α Pβ, so the proof is easy.28

So let cf(α) = ω. The following straightforward argument works for any ccc partial CS iteration where29

are all iterandsQβ are of size≤ ℵ1.30

For notational simplicity we assume Pβ Qβ ⊆ ω1 for all β < α (this is justified by inductive assump-31

tion (5)). By induction, we can assume that for allβ < α there is a denseP∗β⊆ Pβ of sizeℵ1 and that every32

P∗β

is ccc. For eachp ∈ Pα and allβ ∈ dom(p) we can find a maximal antichainApβ⊆ P∗

βsuch that each33

elementa ∈ Apβ

decides the value ofp(β), saya Pβ p(β) = γpβ(a). Writing47 p ∼ q if p ≤ q andq ≤ p, the34

map p 7→ (Apβ, γ

pβ)β∈dom(p) is 1-1 modulo∼. Since eachAp

βis countable, there are onlyℵ1 many possible35

values, therefore there are onlyℵ1 many∼-equivalence classes. Any set of representatives will be dense.36

Alternatively, we can prove (3) directly forP′α. I.e., we can find a≤∗-dense subsetP′′ ⊆ P′α of cardinality37

ℵ1. Note that a conditions (x, p) ∈ P′α essentially depends only onp (cf. Fact 5.11). More specifically, given38

(x, p) we can “transitively48 collapsex aboveα”, resulting in a=∗-equivalent condition (x′, p′). Since39

|α| = ℵ1, there are onlyℵ1ℵ0 = 2ℵ0 many such candidatesx′ and since eachx′ is countable andp′ ∈ x′,40

there are only 2ℵ0 many pairs (x′, p′).41

Proof of (4).42

Ultralaver successor case:Let α = β + 1 with β an ultralaver position. We already know thatPα =43

Pβ ∗Qβ whereQβ is an ultralaver forcing, which in particular is ccc, so by inductionPα is ccc.44

Janus successor case:As above it suffices to show thatQβ, the union of the Janus forcingsQxβ[Hxβ] for45

x ∈ G, is (forced to be) ccc.1

47Since≤ is separative,p ∼ q iff p =∗ q, but this fact is not used here.48In more detail: We define a functionf : Mx → V by induction as follows: Ifβ ∈ Mx ∩ α + 1 or if β = ω2, then f (β) = β.

Otherwise, ifβ ∈ Mx ∩ Ord, then f (β) is the smallest ordinal abovef [β]. If a ∈ Mx \ Ord, then f (a) = f (b) : b ∈ a ∩ Mx. It iseasy to see thatf is an isomorphism fromMx to f [Mx] := Mx′ and thatMx′ is a candidate. Moreover, the ordinals that occur inMx′

are subsets ofα + ω1 together with the interval [ω2, ω2 + ω1]; i.e., there areℵ1 many ordinals that can possibly occur inMx′ , andtherefore there are 2ℵ0 many possible such candidates. Moreover, settingp′ ≔ f (p), it is easy to check that (x, p) =∗ (x, p) (similarlyto Fact 5.11).

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Assume towards a contradiction that this is not the case, i.e., that we have an uncountable antichain2

in Qβ. We already know thatQβ has sizeℵ1 and therefore the uncountable antichain has sizeℵ1. So,3

working inV, we assume towards a contradiction that4

(5.18) x0 R p0 Pβ ãi : i ∈ ω1 is a maximal (uncountable) antichain inQβ.

We construct by induction onn ∈ ω a decreasing sequence of conditions such thatxn+1 satisfies the5

following:6

(i) For all i ∈ ω1 ∩ Mxn there is (inMxn+1) a Pxn+1β

-namea∗i for a Qxn+1β

-condition such that7

xn+1 R p0 Pβ ãi = a∗i .

Why can we get that? Just use Lemma 5.16.8

(ii) If τ is in Mxn a Pxnβ

-name for an element ofQxnβ

, then there isk∗(τ) ∈ ω1 such that9

xn+1 R p0 Pβ (∃i < k∗(τ))ãi ‖Pβ τ.

Also, all thesek∗(τ) are inMxn+1.10

Why can we get that? First note thatxn p0 (∃i ∈ ω1)ãi ‖ τ. SincePβ is ccc,xn forces that there11

is some bound˜k(τ) for i. So it suffices thatxn+1 determines

˜k(τ) to bek∗(τ) (for all the countably12

manyτ).13

Setδ∗ ≔ ω1 ∩⋃

n∈ω Mxn. By Corollary 5.8(4), there is somey such that14

• y ≤ xn for all n ∈ ω,15

• (xn)n∈ω and (a∗i )i∈δ∗ are inMy,16

• (My thinks that)Pyβ

forces thatQyβ

is the union ofQxnβ

.17

Let G beR-generic (overV) containingy, and letHβ bePβ-generic (overV[G]) containingp0.18

SetA∗ ≔ a∗i [Hyβ] : i < δ∗. Note thatA∗ is in My[Hy

β]. We claim19

(5.19) A∗ ⊆ Qyβ[Hyβ] is predense.

Pick anyq0 ∈ Qyβ. So there is somen ∈ ω and someτ which is inMxn a Pxn

β-name of aQxn

β-condition, such20

thatq0 = τ[Hxnβ

]. By (ii) above,xn+1 and thereforey forces (inR) that for somei < k∗(τ) (and therefore21

somei < δ∗) the conditionp0 forces the following (inPβ):22

The conditionsãi andτ are compatible inQβ. Also,

ãi = a∗i andτ both are inQy

β, andQy

β23

is an incompatibility-preserving subforcing ofQβ. ThereforeMy[Hyβ] thinks thata∗i andτ24

are compatible.25

This proves (5.19).26

SinceQyβ[Hyβ] is My[Hy

β]-complete inQβ[Hβ], and sinceA∗ ∈ My[Hy], this implies (as ˙a∗i [H

yβ] =

ãi [G ∗27

Hβ] for all i < δ∗) thatãi [G ∗ Hβ] : i < δ∗ already is predense, a contradiction to (5.18).28

Limit case:We work withP′α, which by definition only contains HCON objects.29

Assume towards a contradiction thatP′α has an uncountable antichain. We already know thatP′α has a30

dense subset of sizeℵ1 (modulo=∗), so the antichain has sizeℵ1.31

Again, work inV. We assume towards a contradiction that32

(5.20) x0 R ãi : i ∈ ω1 is a maximal (uncountable) antichain inP′α.

So eachãi is anR-name for an HCON object (x, p) in V.33

To lighten the notation we will abbreviate elements (x, p) ∈ P′α by p; this is justified by Fact 5.11.34

Fix any HCON objectp andβ < α. We will now define the (R ∗ P′β)-names

˜ι(β, p) and

˜r(β, p): Let G35

beR-generic and containingx0, andH′β

beP′β-generic. LetR be the quotientP′α/H

′β. If p is not inR, set36

˜ι(β, p) =

˜r(β, p) = 0. Otherwise, let

˜ι(β, p) be the minimali such that

ãi ∈ R and

ãi andp are compatible37

(in R), and set˜r(β, p) ∈ R to be a witness of this compatibility. SinceP′

βis (forced to be) ccc, we can38

find (in V[G]) a countable set˜Xι(β, p) ⊆ ω1 containing all possibilities for

˜ι(β, p) and similarly

˜Xr(β, p)39

consisting of HCON objects for˜r(β, p).40

To summarize: For everyβ < α and every HCON objectp, we can define (inV) theR-names˜Xι(β, p)41

and˜Xr(β, p) such that1

(5.21) x0 R P′β

(

p ∈ P′α/H′β → (∃i ∈

˜Xι(β, p)) (∃r ∈

˜Xr(β, p)) r ≤P′α/H′β p,

ãi

)

.

969 revision:2011-05-28

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Similarly to the Janus successor case, we define by inductionon n ∈ ω a decreasing sequence of con-2

ditions such thatxn+1 satisfies the following: For allβ ∈ α ∩ Mxn and p ∈ Pxnα , xn+1 decides

˜Xι(β, p) and3

˜Xr (β, p) to be someXι∗(β, p) andXr∗(β, p). For all i ∈ ω1 ∩ Mxn, xn+1 decides

ãi to be somea∗i ∈ Pxn+1

α .4

Moreover, each suchXι∗ andXr∗ is in Mxn+1, and everyr ∈ Xr∗(β, p) is in Pxn+1α . (For this, we just use5

Fact 5.14 and Lemma 5.15.)6

Setδ∗ ≔ ω1 ∩⋃

n∈ω Mxn, and setA∗ ≔ a∗i : i ∈ δ∗. By Corollary 5.8(4), there is somey such that7

• y ≤ xn for all n ∈ ω,8

• x≔ (xn)n∈ω andA∗ are inMy,9

• (My thinks that)Pyα is defined as the almost FS limit over ¯x.10

We claim thaty forces11

(5.22) A∗ is predense inPyα.

ThenPyα is My-completely embedded intoP′α, and sinceA∗ ∈ My (and since

ãi = a∗i for all i ∈ δ∗) we get12

thatãi : i ∈ δ∗ is predense, a contradiction to (5.20).13

So it remains to show (5.22). LetG beR-generic containingy. Let r be a condition inPyα; we will find14

i < δ∗ such thatr is compatible witha∗i . SincePyα is the almost FS limit over ¯x, there is somen ∈ ω and15

β ∈ α ∩ Mxn such thatr has the formq∧ p with p in Pxnα , q ∈ Py

βandq ≤ pβ.

16Now let H′

βbeP′

β-generic containingq. Work in V[G][H′

β]. Sinceq ≤ pβ, we getp ∈ P′α/H

′β. Let ι∗17

be the evaluation byG ∗ H′β

of˜ι(β, p), and letr∗ be the evaluation of

˜r(β, p). Note thatι∗ < δ∗ andr∗ ∈ Py

α.18

So we know thata∗ι∗ andp are compatible inP′α/H′β

witnessed byr∗. Findq′ ∈ H′β

forcing r∗ ≤P′α/H′β p, a∗ι∗.19

We may findq′ ≤ q. Now q′ ∧ r∗ witnesses thatq∧ p anda∗ι∗ are compatible inPyα.20

Proof of (5).21

This follows from (3) and (4). 22

5.D. The generic alternating iteration P. In Lemma 5.17 we have seen:23

Corollary 5.23. Let G beR-generic. Then we can construct49 (in V[G]) an alternating iterationP such24

that the following holds:25

• P is ccc.26

• If x ∈ G, then x canonically embeds intoP. (In particular, a Pω2-generic filter Hω2 induces a27

Pxω2

-generic filter over Mx, called Hxω2

.)28

• EachQα is the union of all Qxα[H

xα] with x ∈ G.29

• Pω2 is equivalent to the direct limitP′ω2of G: There is a dense embedding j: P′ω2

→ Pω2, and for30

each x∈ G the function p7→ j(x, p) is the canonical embedding.31

Lemma 5.24. Let x∈ R. ThenR forces the following: x∈ G iff x canonically embeds intoPω2.32

Proof. If x ∈ G, then we already know thatx canonically embeds intoP.33

So assume (towards a contradiction) thaty forces thatx embeds, and thatx ⊥ y (i.e.,y x < G). Work34

in V[G] wherey ∈ G. Bothx (by assumption) andy ∈ G canonically embed intoP. Let N be an elementary35

submodel ofHV[G](χ∗) containingx, y, P; let z = (Mz, Pz) be the ord-collapse of (N, P). Thenz ∈ V (asR36

is σ-closed) andz ∈ R, and (by elementarity)z≤ x, y. This shows thatx ‖R y, i.e.,y cannot forcex < G, a37

contradiction. 38

Using ccc, we can now prove a lemma that is in fact stronger than the lemmas in the previous section 5.C:39

Lemma 5.25. The following is forced byR: Let N ≺ HV[G] (χ∗) be countable, and let y be the ord-collapse40

of (N, P). Then y∈ G. Moreover, if x∈ G∩ N, then y≤ x.41

Proof. Work in V[G] with x ∈ G. Pick an elementary submodelN containingx andP. Let y be the ord-42

collapse of (N, P) via a collapsing mapk. As above, it is clear thaty ∈ R andy ≤ x. To showy ∈ G, it43

is (by the previous lemma) enough to show thaty canonically embeds. We claim thatk−1 is the canonical44

embedding ofy into P. The crucial point is to showMy-completeness. LetB ∈ My be a maximal antichain45

of Pyω2

, sayB = k(A) whereA ∈ N is a maximal antichain ofPω2. So (by ccc)A is countable, henceA ⊆ N.46

So not onlyA = k−1(B) but evenA = k−1[B]. Hencek−1 is anMy-complete embedding. 1

49in an “absolute way”: GivenG, we first defineP′ω2to be the direct limit ofG, and then inductively constructPα from P′ω2

.

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Remark 5.26. Using this lemma we can easily reprove the countable chain condition ofPω2 and in fact all2

other lemmas in section 5.C. We will do this for Fact 5.14 in Corollary 6.4.3

6. T BC+dBC4

We first prove that no uncountableX in V will be smz or sm in the extension.50 Then we show how to5

modify the argument to work for all uncountable sets inV[G ∗ H].6

6.A. BC+dBC for ground model sets. A schematic diagram illustrating the arguments in this section can7

be found in Figure 2 on page 8.8

Lemma 6.1. Let X∈ V be an uncountable set of reals. ThenR ∗ Pω2 forces that X is not smz.9

Proof.10

(1) Fix any evenα < ω2 (i.e., an ultralaver position) in our iteration. The ultralaver forcingQα adds a11

(canonically defined code for a) closed null setF constructed from the ultralaver realℓα. (Recall12

the ultralaver wish list 2.1, item (2) and item (3). In the following, when we consider various13

forcingsQα, Qα, Qxα, we treatF not as an actual name, but rather as a definition which dependson14

the forcing used.)15

(2) According to Theorem 1.2, it is enough to show thatX+ F is non-null in theR ∗Pω2-extension, or16

equivalently, in everyR∗Pβ-extension (α < β < ω2). So assume towards a contradiction that there17

is aβ > α and anR ∗ Pβ-name˜Z of a (code for a) Borel null set such that some (x, p) ∈ R ∗ Pω218

forces thatX + F ⊆˜Z.19

(3) Using the dense embeddingjω2 : P′ω2→ Pω2, we may replace (x, p) by a condition (x, p′) ∈ R∗P′ω2

.20

According to Fact 5.14 (recall that we now know thatPω2 satisfies ccc) and Lemma 5.15 we can21

assume thatp′ is already aPxβ-conditionpx and that

˜Z is (forced byx to be the same as) aPx

β-name22

Zx in Mx.23

(4) We construct (inV) an iterationP in the following way:24

(a) Up toα, we take an arbitrary alternating iteration into whichx embeds. In particular,Pα will25

be proper and hence forces thatX is still uncountable.26

(b) Let Qα be any ultralaver forcing (overQxα in caseα ∈ Mx). So according to item (2) of the27

ultralaver wish list 2.1, we know thatQα forces thatX + F is not null.28

Therefore we can pick (inV[Hα+1]) some ˙r in X + F which is random over (the countable29

model)Mx[Hxα+1], whereHx

α+1 is induced byHα+1.30

(c) In the rest of the construction, we preserve randomness of r overMx[Hxζ] for eachζ ≤ ω2. We31

can do this using an almost CS iteration overx where at each Janus position we use a random32

version of Janus forcing and at each ultralaver position we use a suitable ultralaver forcing;33

this is possible by Lemma 4.32. By Lemma 4.34, this iterationwill preserve the randomness34

of r.35

(d) So we getP overx (with canonical embeddingix) andq ≤Pω2ix(px) such thatqβ forces (in36

Pβ) that r is random overMx[Hxβ], in particular that ˙r < Zx.

37

We now pick a countableN ≺ H(χ∗) containing everything and ord-collapse (N, P) to y ≤ x. (See38

Fact 5.4.) SetXy≔ X ∩ My (the image ofX under the collapse). By elementarity,My thinks that39

(a)–(d) above holds forPy and thatXy is uncountable. Note thatXy ⊆ X.40

(5) This gives a contradiction in the obvious way: LetG beR-generic overV and containy, and letHβ41

bePβ-generic overV[G] and containqβ. So My[Hyβ] thinks thatr < Zx (which is absolute) and42

thatr = x+ f for somex ∈ Xy ⊆ X and f ∈ F (actually even inF as evaluated inMy[Hyα+1]). So43

in V[G][Hβ], r is the sum of an element ofX and an element ofF. So (y, q) ≤ (x, p′) forces that44

r ∈ X + F \ Z, a contradiction to (2). 45

Of course, we need this result not just for ground model setsX, but forR ∗ Pω2-names˜X = (

˜xi : i ∈ ω1)46

of uncountable sets. It is easy to see that it is enough to dealwith R ∗ Pβ-names for (all)β < ω2. So given47

˜X, we can (in the proof) pickα such that

˜X is actually anR∗Pα-name. We can try to repeat the same proof;1

50Note that for this weak version, it would be enough to producea generic iteration of length 2 only, i.e.,Q0 ∗Q1, whereQ0 is anultralaver forcing andQ1 an corresponding Janus forcing.

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however, the problem is the following: When constructingP in (4), it is not clear how to simultaneously2

make all the uncountably many names (˜xi) into P-names in a sufficiently “absolute” way. In other words:3

It is not clear how to end up with someMy and Xy uncountable inMy such that it is guaranteed thatXy4

(evaluated inMy[Hyα]) will be a subset of

˜X (evaluated inV[G][Hα]). We will solve this problem in the5

next section by factoringR.6

Let us now give the proof of the corresponding weak version ofdBC:7

Lemma 6.2. Let X∈ V be an uncountable set of reals. ThenR ∗ Pω2 forces that X is not strongly meager.8

Proof. The proof is parallel to the previous one:9

(1) Fix any evenα < ω2 (i.e., an ultralaver position) in our iteration. The Janus forcingQα+1 adds a10

(canonically defined code for a) null setZ∇. (Recall the Janus wish list 3.1, item (1).)11

(2) According to (1.8), it is enough to show thatX+ Z∇ = 2ω in theR ∗Pω2-extension, or equivalently,12

in everyR ∗ Pβ-extension (α < β < ω2). (For every realr, the statementr ∈ X + Z∇, i.e.,13

(∃x ∈ X) x+ r ∈ Z∇, is absolute.) So assume towards a contradiction that thereis aβ > α and an14

R ∗ Pβ-name˜r of a real such that some (x, p) ∈ R ∗ Pω2 forces that

˜r < X + Z∇.15

(3) Again, we can assume that˜r is aPx

β-name ˙r x in Mx.

16

(4) We construct (inV) an iterationP in the following way:17

(a) Up toα, we take an arbitrary alternating iteration into whichx embeds. In particular,Pα again18

forces thatX is still uncountable.19

(b1) LetQα be any ultralaver forcing (overQxα). ThenQα forces thatX is not thin (see item (4) of20

the ultralaver wish list 2.1).21

(b2) Let Qα+1 be a countable Janus forcing. SoQα+1 forcesX + Z∇ = 2ω. (See Janus wish22

list 3.1(1).)23

(c) We continue the iteration in aσ-centered way. I.e., we use an almost FS iteration overx24

of ultralaver forcings and countable Janus forcings, usingtrivial Qζ for all ζ < Mx; see25

Lemma 4.17.26

(d) SoPβ still forces thatX+ Z∇ = 2ω, and in particular that ˙r x ∈ X+ Z∇. (See Janus wish list 3.127

item (1)).28

Again, by collapsing someN as in the previous proof, we gety ≤ x andXy ⊆ X.29

(5) This again gives the obvious contradiction: LetG beR-generic overV and containy, and letHβ30

bePβ-generic overV[G] and containp. SoMy[Hyβ] thinks thatr = x+ z for somex ∈ Xy ⊆ X and31

z ∈ Z∇ (this time,Z∇ is evaluated inMy[Hyβ]), contradicting (2).

32

6.B. A factor lemma. We can restrictR to anyα∗ < ω2 in the obvious way: Conditions are pairsx =33

(Mx, Px) of nice candidatesMx (containingα∗) and alternating iterationsPx, but nowMx thinks thatPx has34

lengthα∗ (and notω2). We call this variantRα∗.35

Note that all results of Section 5 aboutR are still true forRα∗. In particular, wheneverG ⊆ Rα∗ is36

generic, it will define a direct limit (which we callP′∗), and an alternating iteration of lengthα∗ (calledP∗);37

again we will have thatx ∈ G iff x canonically embeds intoP∗.38

There is a natural projection map fromR (more exactly: from the dense subset of thosex which satisfy39

α∗ ∈ Mx) into Rα∗, mappingx = (Mx, Px) to xα∗ ≔ (Mx, Pxα∗). (It is obvious that this projection is40

dense and preserves≤.)41

There is also a natural embeddingϕ from Rα∗ to R: We can just continue an alternating iteration of42

lengthα∗ by appending trivial forcings.43

ϕ is complete: It preserves≤ and⊥. (Assume thatz≤ ϕ(x), ϕ(y). Thenzα∗ ≤ x, y.) Also, the projection44

is a reduction: Ify ≤ xα∗ in Rα∗, then letMz be a model containg bothx andy. In Mz, we can first45

construct an alternating iteration of lengthα∗ overy (using almost FS overy, or almost CS — this does46

not matter here). We then continue this iterationPz using almost FS or almost CS overx. Sox andy both47

embed intoPz, hencez= (Mz, Pz) ≤ x, y.48

So according to the general factor lemma of forcing theory, we know thatR is forcing equivalent to49

Rα∗ ∗ (R/Rα∗), whereR/Rα∗ is the quotient ofR andRα∗, i.e., the (Rα∗-name for the) set ofx ∈ R50

which are compatible (inR) with all ϕ(y) for y ∈ Gα∗ (the generic filter forRα∗), or equivalently, the set51

of x ∈ R such thatxα∗ ∈ Gα∗. So Lemma 5.24 (relativized toRα∗) implies:1

(6.3) R/Rα∗ is the set ofx ∈ R that canonically embed (up toα∗) into Pα∗ .

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Setup. Fix someα∗ < ω2 of uncountable cofinality.51 Let Gα∗ beRα∗-generic over V and work in2

V∗ ≔ V[Gα∗]. SetP∗ = (Pβ)β<α∗ , the generic alternating iteration added byRα∗. LetR∗ be the quotient3

R/Rα∗.4

We claim thatR∗ satisfies (inV∗) all the properties that we proved in Section 5 forR (in V), with the5

obvious modifications. In particular:6

(A)α∗ R∗ is ℵ2-cc, since it is the quotient of anℵ2-cc forcing.7

(B)α∗ R∗ does not add new reals (and more generally, no new HCON objects), since it is the quotient of a8

σ-closed forcing.529

(C)α∗ Let G∗ beR∗-generic overV∗. ThenG∗ is R-generic overV, and therefore Corollary 5.23 holds10

for G∗. (Note thatP′ω2and thenPω2 is constructed fromG∗.) Moreover, it is easy to see53 that P11

starts withP∗.12

(D)α∗ In particular, we get a variant of Lemma 5.25: The following is forced byR∗: Let N ≺ HV[G∗ ](χ∗)13

be countable, and lety be the ord-collapse of (N, P). Theny ∈ G∗. Moreover: Ifx ∈ G∗ ∩ N, then14

y ≤ x.15

We can use the last item to prove theR∗-version of Fact 5.14 (which we could of course also reprove16

directly):17

Corollary 6.4. Assume that x∈ R∗ forces that p∈ Pω2. Then there is a y≤ x and a py ∈ Pyω2

such that y18

forces py =∗ p.19

Proof. Let G∗ containx. In V[G∗], pick an elementary submodelN containingx, p, P and let (Mz, Pz, pz)20

be the ord-collapse of (N, P, p). Thenz ∈ G∗. This whole situation is forced by somey ≤ z≤ x ∈ G∗. Soy21

andpy is as required, wherepy ∈ Pyω2

is the canonical image ofpz. 22

We now claim thatR∗Pω2 forces BC+dBC. We know thatR is forcing equivalent toRα∗∗R∗. Obviously23

we have24

R ∗ Pω2 = Rα∗ ∗ R∗ ∗ Pα∗ ∗ Pα∗ , ω2

(wherePα∗ , ω2 is the quotient ofPω2 andPα∗ ). Note thatPα∗ is already determined byRα∗, soR∗ ∗ Pα∗ is25

(forced byRα∗ to be) a productR∗ × Pα∗ = Pα∗ × R∗.26

But note that this is not the samePα∗ ∗ R∗, where we evaluate the definition ofR∗ in thePα∗ -extension27

of V[Gα∗]: We would get new candidates and therefore new conditions in R∗ after forcing withPα∗ . In28

other words, we can unfortunatelynot just argue as follows:29

Wrong argument. R ∗ Pω2 is the same as (Rα∗ ∗ Pα∗ ) ∗ (R∗ ∗ Pα∗ ,ω2); so given anR ∗ Pω2-nameX of a30

set of reals of sizeℵ1, we can chooseα∗ large enough so thatX is an (Rα∗ ∗Pα∗ )-name. Then, working in31

the (Rα∗ ∗ Pα∗ )-extension, we just apply Lemmas 6.1 and 6.2.32

So what do we do instead? Assume that˜X =

˜ξi : i ∈ ω1 is anR ∗ Pω2-name for a set of reals of33

sizeℵ1. So there is aβ < ω2 such that˜X is added byR ∗ Pβ (usingℵ2-cc of R). In theR-extension,34

Pβ is ccc, therefore we can assume that each˜ξi is a system of countably many countable antichains

Ãm

i35

of Pβ, together with functions˜f mi :

Ãm

i → 0, 1. For the following argument, we prefer to work with the36

equivalentP′β

instead ofPβ. We can assume that each of the sequencesBi ≔ (Ãm

i ,˜f mi )m∈ω is an element37

of V (sinceP′β

is a subset ofV and sinceR is σ-closed). So eachBi is decided by a maximal antichainZi38

of R. SinceR is ℵ2-cc, theseℵ1 many antichains all are contained in someRα∗ with α∗ ≥ β.39

From now on, we work in theRα∗-extensionV∗. So inV∗ we have the following situation: Eachξi is40

a very “absolute”R∗ ∗ Pα∗ -name (or equivalently,R∗ × Pα∗ -name), in fact they are already determined by41

antichains that are inPα∗ and do not depend onR∗. So we can interpret them asPα∗ -names.42

Note that theξi are forced (byR∗ ∗ Pα∗ ) to be pairwise different, and therefore already byPα∗ .1

51Probably the cofinality is completely irrelevant, but the picture is clearer this way.52It is easy to see thatR∗ is evenσ-closed, by “relativizing” the proof forR, but we will not need this.53Let P′∗

βbe the direct limit ofGα∗ (for β ≤ α∗), andP′

βthe direct limit ofG∗. The functionkβ : P′∗

β→ P′

βthat maps (x, p) to

(ϕ(x), p) preserves≤ and⊥ and is surjective modulo=∗, see Fact 5.11(3). So it is clear that definingP∗ by induction fromP′∗ω2yields

the same result as definingP from P′ω2.

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So now we can just repeat the proofs of BC and dBC of Section 6.A, with the following modifications2

(the modifications are the same for both proofs): We fix the uncountable set˜X, choose the accordingα∗ as3

above, and work in theRα∗ extensionV∗.4

(1) Instead of any ultralaver positionα < ω2, we obviously have to choose anα ≥ α∗.5

(2) No change here (of course we now have anR∗ ∗ Pβ-name).6

(3) Here, we use Corollary 6.4.7

(4) The iteration obviously has to start withP∗ (which is ccc), the rest is the same. SinceP starts with8

P∗, the ord-collapsey of (N, P) is in R∗. Now comes the relevant point of the factoring: Since9

˜X consists of “absolute”P∗α∗ -names, we know that ify is in theR∗-generic filterG∗, and if H is10

Pω2-generic overV[G∗], then Xy[H] = X[H] ∩ My ⊆ X[H]. So y thinks thatXy is a name for an11

uncountable set, and we know that thisPyω2

-name will be evaluated to a subset of˜X.12

(5) No change here.13

7. A 14

The following is not needed for understanding the paper, we just briefly comment on alternative ways15

some notions could be defined.16

7.A. Regarding “alternating iterations”. We call the set ofα ∈ ω2 such thatQα is (forced to be) nontriv-17

ial the “true domain” of P (we use this notation in this remark only). ObviouslyP is naturally isomorphic18

to an iteration whose length is the order type of its true domain. In Definitions 5.1 and 5.3, we could have19

imposed the the following additional requirements. All these variants lead to equivalent forcing notions.20

(1) Mx is (an ord-collapse of) anelementarysubmodel ofH(χ∗).21

This is equivalent, as conditions coming from elementary submodels are dense in ourR, by22

Fact 5.4.23

While this definition looks much simpler and therefore nicer(we could replace ord-transitive mod-24

els by the better understood elementary models), it would not make things easier and just “hides”25

the point of the construction: For example, we use modelsMx that are (an ord-collapse of) an26

elementary submodel ofHV′(χ∗) for some forcing extensionV′ of V.27

(2) Require that (Mx thinks that) the true domain ofPx isω2.28

This is equivalent for the same reason as (1) (and this requirement is compatible with (1)).29

This definition would allow to drop the “trivial” option fromthe definition, but it would make the30

dBC argument more cumbersome, as an iteration with uncountably many nontrivial iterandsQα is31

notσ-centered.32

(3) Alternatively, require that (Mx thinks that) the true domain ofPx is countable.33

Again, equivalence can be seen as in (1), again (3) is compatible with (1) but obviously not with (2).34

This requirement would not make the definition easier, so there is no reason to adopt it. It would35

have the slight inconvenience that instead of using ord-collapses as in Fact 5.4, we would have to36

put another model on top to make the iteration countable. Also, it would have the (purely aesthetic)37

disadvantage that the generic iteration itself does not satisfy this requirement.38

(4) Also, we could have dropped the requirement that the iteration is proper. It is never directly used,39

and “densely”P is proper anyway. (E.g., in Lemma 6.1(4a), we would just construct P up toα to40

be proper or even ccc, so thatX remains uncountable.)41

7.B. Regarding “almost CS iterations and separative iterands”. Recall that in Definition 4.6 we required42

that each iterandQα in a partial CS iteration is separative. This implies the property (actually: the three43

equivalent properties) from Fact 4.8. Let us call this property “suitability” for now. Suitability is a prop-44

erty of the limit Pε of P. Suitability always holds for finite support iterations andfor countable support45

iterations. However, if we do not assume that eachQα is separative, then suitability may fail for par-46

tial CS iterations. We could drop the separativity assumption, and instead add suitability as an additional47

requirement to the definition of partial CS limit.48

The disadvantage of this approach is that we would have to check in all constructions of partial CS49

iterations that suitability is indeed satisfied (which is straightforward but surprisingly is rather cumbersome,50

in particular in the case of the almost CS iteration, where anadditional, stronger form of suitability has to51

be introduced).1

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In contrast, the disadvantage of assuming thatQα is separative is minimal and purely cosmetic: It is2

well known that every quasiorderQ can be made into a separative one which is forcing equivalentto the3

originalQ (e.g., by just redefining the order to be≤∗Q).4

7.C. Regarding “preservation of random and quick sequences”. Recall Definition 4.31 of local preser-5

vation of random reals and Lemma 4.32.6

In some respect the dense setsDn are unnecessary. For ultralaver forcingLD, the notion of a “quick”7

sequence refers to the setsDn of conditions with stem of length at leastn.8

We could define a new partial order onLD as follows:9

q ≤′ p ⇔ (q = p) or (q ≤ p and the stem ofq is strictly longer than the stem ofp)

Then (LD,≤) and (LD,≤′) are forcing equivalent, and any≤′-interpretation of a new real will automatically10

be quick.11

Note however that (LD,≤′) is now not separative any more. Therefore we chose not to take this approach,12

since losing separativity causes technical inconvenience, as described in 7.B.13

R14

[BJ95] Tomek Bartoszynski and Haim Judah.Set theory. A K Peters Ltd., Wellesley, MA, 1995. On the structure of thereal line.15

[Bor19] E. Borel. Sur la classification des ensembles de mesure nulle.Bull. Soc. Math. France, 47:97–125, 1919.16

[BS03] Tomek Bartoszynski and Saharon Shelah. Strongly meager sets of size continuum.Arch. Math. Logic, 42(8):769–779,17

2003.18

[BS10] Tomek Bartoszynski and Saharon Shelah. Dual Borel conjecture and Cohen reals.J. Symbolic Logic, 75(4):1293–1310,19

2010.20

[Car93] Timothy J. Carlson. Strong measure zero and strongly meager sets.Proc. Amer. Math. Soc., 118(2):577–586, 1993.21

[GK06] Martin Goldstern and Jakob Kellner. New reals: can live with them, can live without them.MLQ Math. Log. Q., 52(2):115–22

124, 2006.23

[GMS73] Fred Galvin, Jan Mycielski, and Robert M. Solovay. Strong measure zero sets.Notices of the AMS, pages A–280, 1973.24

[Gol93] Martin Goldstern. Tools for your forcing construction. In Set theory of the reals (Ramat Gan, 1991), volume 6 ofIsrael25

Math. Conf. Proc., pages 305–360. Bar-Ilan Univ., Ramat Gan, 1993.26

[Jec03] Thomas Jech.Set theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. The third millennium27

edition, revised and expanded.28

[JS90] Haim Judah and Saharon Shelah. The Kunen-Miller chart (Lebesgue measure, the Baire property, Laver reals and preser-29

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A45

E-mail address: [email protected]

URL: http://www.tuwien.ac.at/goldstern/47

KG R C M L, UW, W Sß 25, 1090 W, A48


URL: http://www.logic.univie.ac.at/∼kellner/50

E I M, E J. S C, G R, T H U J, J,51

91904, I, D M, R U, N B, NJ 08854, USA52


URL: http://shelah.logic.at/54

I D M G, T UW, W Hß 8–10/104, 1040 W,55

A1

http://arxiv.org/abs/0910.2132

969 revision:2011-05-28

modified:2011-05-29



URL: http://www.wohofsky.eu/math/2033

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Martin Goldstern, Jakob Kellner, Saharon Shelah and Wolfgang Wohofsky- Borel Conjecture and Dual...

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