The Axiom of Infinity

8/4/2019 The Axiom of Infinity

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The Axiom of Infinity and Transformations j: V VAuthor(s): Paul CorazzaSource: The Bulletin of Symbolic Logic, Vol. 16, No. 1 (Mar., 2010), pp. 37-84Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/25614509 .Accessed: 13/06/2011 17:54

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The Bulletin of Symbolic LogicVolume 16, umber 1, arch 2010

THE AXIOM OF INFINITYAND TRANSFORMATIONS jr: V -+ V

PAUL CORAZZA

Abstract. We suggest a new approach for addressing the problem of establishing an ax

iomatic foundation for large cardinals. An axiom asserting the existence of a large cardinal

can naturally be viewed as a strong Axiom of Infinity. However, it has not been clear on the

basis of our knowledge of co itself, or of generally agreed upon intuitions about the true nature

of the mathematical universe, what the right strengthening of the Axiom of Infinity is?which

large cardinals ought to be derivable? It was shown in the 1960s by Lawvere that the existenceof an infinite set is equivalent to the existence of a certain kind of structure-preserving trans

formation from V to itself, not isomorphic to the identity. We use Lawvere's transformation,rather than w, as a starting point for a reasonably natural sequence of strengthenings and

refinements, leading to a proposed strong Axiom of Infinity. A first refinement was discussed

in later work by Trnkova-Blass, showing that if the preservation properties of Lawvere's

tranformation are strengthened to the point of requiring it to be an exact functor, such a

transformation is provably equivalent to the existence of a measurable cardinal. We proposeto push the preservation properties as far as possible, short of inconsistency. The resultingtransformation V ? V is strong enough to account for virtually all large cardinals, but is at

the same time a natural generalization of an assertion about transformations V ? V knownto be equivalent to the Axiom of Infinity.

?0. Introduction. What is the right notion of "infinite" inmathematics?Prior to the work of Cantor, the only acceptable notion was "potential infin

ity": arbitrarily argefinite izes were allowed, but all such finite sizes couldnot be collected together to form a single, well-defined set. Both because of

philosophicalissues and

seemingmathematical

paradoxes,actual infinities

were not considered legitimate objects of mathematical investigation. Cantor convinced the mathematical world that mathematics without an actualinfinite s too impoverished. (See [15], [27], [28] for a discussion of these

points.)Part of Cantor's legacy is the presence of theAxiom of Infinity n the

standard set theory FC. The Axiom of Infinity sserts that n infinite et

Received April 24, 2008.

2010Mathematics Subject Classification. Primary 03E55; Secondary 03E40.Key words and phrases. Axiom of Infinity, A, Wholeness Axiom, large cardinal, exact

functor, critical point, Lawvere, universal element.

The results of this paper were presented at the 2008Annual Meeting of theAssociationfor Symbolic Logic, March 27-30, Irvine, California

? 2010, Association for Symbolic Logic1079-8986/10/1601 -0002/$5.80

37


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38 PAULCORAZZA

exists, and this axiom, in conjunction with the other ZFC axioms, sufficesto build Cantor's entire theory of transfinite cardinals.

Ironically, even as ZFC itself as crystallizing nto its final form as the

de facto universal foundation for mathematics, the first large cardinals werebeing discovered: Hausdorff discovered weakly inaccessible cardinals [18]in 1908and Mahlo discovered (weakly)Mahlo cardinals [29] in 1911. As

Godel would later show, large cardinals are one type of mathematical entitythat is underivable from ZFC.

It was clear even inHausdorfFs time that large cardinals represented new,

stronger otions of infinity, iving rise to natural models of set theory see[20, Introduction]). Cantor gave us a "minimal" notion of actual infinite,but

perhapsa

stronger ypeof

infinityan

trulye asserted to exist in the

universe. How strong should the Axiom of Infinity be?Since the time of Godel's Second Incompleteness Theorem, which demon

strated among other things that large cardinals are not derivable from ZFC,the attitude in the community of set theorists toward large cardinals hasbeen mixed.1 Some have been unconcerned with existence of large cardinals.Others have actively attempted to prove that some or all large cardinals arein fact inconsistent.2 And a third group attempted to devise heuristics, often

based on Cantor's vision of the structure of the universe, to legitimize the

presence of large cardinals in the universe. We consider two examples of suchheuristics from this relatively arly period in large cardinal history; both ofthese heuristics were introduced by Godel and developed further by others

(see 21]):One of Cantor's perspectives about the mathematical universe is that the

transition from finite to transfinite is reasonably smooth and consequently,the universe is fairly homogenous ([15, Section 1.3], [20, Introduction]).From this point of view, itwould be too "accidental" for co to be the only

cardinal having the large cardinal properties of (for xample) inaccessibilityand measurability, and so we conclude that there are inaccessible and measurable cardinals in the universe. This is an example of the generalizationheuristic.

A second insight f Cantor's is that the Absolute Infinite sbeyond mathematical determination. Neither the universe itself nor the class ON of ordinals should be able to be captured by a singlefirst-order roperty [15,Section 1.3], [20, Introduction]. Therefore, if it can be legitimately laimedthat V or ON satisfies such a

property,some actual set or ordinal must also

satisfy heproperty. his is n intuitive eflection rinciple', n [31], einhardtattempts to formalize this type of reflection. It can be argued on intuitive

^ee [12]for survey f a variety of contemporary views on these issues.2A striking xample of such an effort hat ore great fruits as Silver's early efforts oprovethe inconsistency fmeasurable cardinals; his efforts esulted inmany of the foundationalresults on 0#.


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V V 39

grounds that ON is inaccessible, Mahlo, and perhaps much more, and so bythe reflection principle, at least inaccessibles and Mahlos should exist in theuniverse. See the work by Paul Bernays [1]where arguments of this kind are

motivatedand

developedmuch

further.For the most part, while these early heuristics may have been useful in justifying ome of the smaller large cardinals, they fail tomotivate the strongerand often more complicated large cardinals that are widely used today, suchas Woodin, strong, superstrong and supercompact cardinals. (See, however, [27], [28]for an excellent survey of heuristics for ustifying ven thestrongest xioms.) Yet, despite the fact that completely satisfactory impleheuristics have not been found for these stronger types of infinities, there iswider acceptance than ever of large cardinals in the universe. Though this

development may be due partly to long-time familiarity with the concepts, itismost likely due to the deep connections that were discovered byWoodin,Steel, Martin, Kunen, and others, between some of the very strongest largestcardinals and sets of small cardinality, particularly sets of reals. One of the

pioneering theorems in this domain, due toMartin [30], established determi

nacy of all 11^sets of reals (from hich it follows, for example, that ll suchsets of reals are Lebesgue measurable and have the property of Baire), as

suming the existence f an I2embedding : Vx?> Vx. Inunpublished work,Woodin extended the technique to show that, assuming an In embeddingj: L(F^+i)

?L(^+i), all sets of reals belonging toL(R) are determined.

In a different direction, Kunen [23] showed how to collapse a huge cardinal to obtain in a forcing extension an (^-saturated ideal on co\. All theseresults were shown later to require much weaker large cardinal hypotheses,but the rich structural connections between these strong axioms and sets be

longing to the earlier stages of the universe strengthened confidence in eventhese strongest arge cardinal notions. (See [20,Chapter 31] for a surveyof these results.) Reflecting similar sentiments, in his 2008 Godel Lecture,3H. Woodin expressed his belief that the "hierarchy of large cardinal axiomshas emerged as an intrinsic part of set theory."

Despite growing acceptance of large cardinals, more clear than ever is thefact that there is no axiomatic foundation for their study. ZFC has provenitself to be a nearly universal foundation for the rest of mathematics; itwould

be only natural to expect that an extension of ZFC should emerge that wouldprovide a foundation for all of mathematics, including large cardinals. Largecardinals have been studied long enough and with enough rigor to warrant a

theory imilar n spirit operhaps the theory f groups or topological spaces.Although the study of large cardinals continues on without such a formaltheory, there are questions one simply cannot ask without such a theory.

To make this point clearer, we consider here an example of such a question

Presented at the 2008Annual Meeting of theAssocialtion for Symbolic Logic, in Irvine,California.


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40 PAULCORAZZA

and discuss later in the paper how a well-formed theory could attempt toanswer it:

Test Question. Which large cardinals have their own brand of

Laver sequence?Recall that Laver sequence (originally efined by Laver [24]) is function

f\ k ?> VK such that for any x and any a > max(/c, |TC(x)|), there is a su

percompact ultrafilter on PKXsuch that if u is the canonical embedding,thenyV(/)(*) = x.

In order to address the Test Question, certain points need to be clarified.

First, one needs to have a notion of Laver sequence that could apply to other

types of large cardinals. Laver sequences for strong and extendible cardinals

have been defined nd proven to exist [3], [4], [14].But the question cannotreally be answered in general without knowing which large cardinals existin the first place, and then, beyond mere existence, some form for candidate

large cardinals that would permit a general definition of "Laver sequence."In earlier work on the Test Question [4],we attempted to address these

issues. We used as our foundational axiom the axiom (schema) discussed inParts 1and 2 of this aper: theWholeness Axiom (WA).We also provided ageneral definition of a broad class of large cardinals that would be candidatesfor

admittingsome form of Laver

sequence;the

backgroundWholeness

Axiom ensured the existence of the classes defined in this way. In Part 1we

will review some of the results and point out some of the added value thatresulted from having a strong axiomatic framework inwhich to work.

In devising an appropriate extension of ZFC to account for large cardinals,it is of course necessary to decide which large cardinals should be declared to"exist." From the trends we see in set theory research and Woodin's recent

remarks, it isnot perhaps too bold to seek an axiomatic foundation for all the

large cardinals?not known to be inconsistent?that have been studied over

the years. Rather than viewing such an axiom system as a commitment to abelief system, it could be viewed as meeting the practical need of providinga useful robust framework for studying the things that are already beingstudied.4 Even ifwe agree to admit all of the better known large cardinals,it is still necessary to decide which sorts of axiom candidates ought to betaken seriously; newould expect that good axiomwould provide not onlythe intended consequences, but in some way be "natural" enough to belongamong the foundational axioms for all of mathematics. In this paper, we

will offer one such axiom and make a case for its naturalness.

4An interesting arallel to this point of view is the fact that some set theorists are not

personally convinced of the consistency even of ZFC itself; ome for example have arguedthat Z?-Replacement for n > 2 is unconvincing. (Solovay discusses this point of view

briefly nhis FOM post, August 17,2007.) Nevertheless, these set theorists freely se ZFCwithout meticulous concern over the degree of Replacement involved in their rguments. Afoundation for all large cardinals could be viewed in a similar fashion.


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42 PAULCORAZZA

In its original form, Lawvere's Theorem talked about cartesian closed

categories, rather than models of ZFC-

Infinity. A cartesian closed categoryisone that has all finite roducts (for ets, these re just cartesian products)and, for any pair of objects A,B, has an exponential object AB, much as wehave in amodel of ZFC or ZFC - Infinity. s we discuss further nPart 2,any model of ZFC

?Infinity can be viewed as a cartesian closed category,

so the version of Lawvere's theorem given here is an immediate consequenceof his original.

Quantification over classes that appears to occur in the statement of thetheorem can be eliminated in the usual ways. In the proof of (1) (2),Fand G are given explicitly s classmaps. For (2) => (1), the result can beviewed as a schema of

theorems,one for each F. In either direction,

jis

simply the composition G oF.We observe that in the theorem, he actors of are endowed with strong

preservation properties. For instance F preserves all colimits; examples ofcolimits in a category of sets are coproducts (which are disjoint unions)and direct limits f directed systems. Dually, G preserves all limits; limitsinclude products (in the usual set-theoretic sense) and inverse limits. Theseare very strong preservation properties in the context of category theory, andarise because of the adjoint relationship between F and G. The class K?is the category of endos (endomorphisms) A ?> A, where A is a set in V.

The statement F H G, where F: C V and G: V?

C, means that for

every object a of C and every b inV, the arrows from (a) to b are in 1-1correspondence with those from to G(b), and the bijection isnatural ina and b. A familiar example of such an adjunction is found when C is thecategory of sets and V is the category of vector spaces over a fixed field.

Then, if (X) is the vector space freely enerated by X, and G{V) is theset V without vector space structure, it follows that F H G and G is the

forgetful functor, a situation exactly parallel to that in Lawvere's Theorem.Detailed definitions are provided in Part 2. We will henceforth call thefunctor :V ?? V defined inpart (2) of the theorem the awvere functor.

The proof of existence of j from theAxiom of Infinity nLawvere's Theorem requires one to define a left djoint F to the already defined forgetfulfunctor G. A definition fF on objects isgiven by

F{A) = A x co-> A x co,

where succ is the successor function on co. Defining F on arrows is straightforward. As we indicate inmore detail in Part 2, a category-theoretic version

of definition y recursion derivable from theAxiom of Infinity) an be usedto establish that HG.

Conversely, to establish theAxiom of Infinity rom the existence of j,one can use the familiar trick f examining the critical point of j. Arguing


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V V 43

category-theoretically, one shows that y (0)= 0 and = co, so 1 is the

critical point of themapping, and its mage explicitly ives us an infinite et.Lawvere's result is striking because it implies that to understand the Axiom

of Infinity, hich asserts simply hat certain set exists (suchas co),onemustcome to terms with tranformational dynamics of the universe as a whole. Infact, it tells us something that we are not perhaps accustomed to believing:that the universe comes equipped with an auxiliary map j: V ?> V that has

strong structure-preserving characteristics.5Lawvere's Theorem opens the door to generalizing to much stronger no

tions of infinity, in a way that the usual Axiom of Infinity does not. Recallthat we are searching for the "right" version of the Axiom of Infinity. Whenlimited to the usual version, the

possible generalizationsdo not go very far.

When instead we ask, "What is the right version of Lawvere's Axiom of In

finity?" we see new possibilities. The task then becomes one of formulatingthe preservation properties of j as clearly as possible and then strengtheningthem as much as possible. In Lawvere's Theorem, the strong preservationproperties belong not to j itself, ut to its factors F and G. So our firstattempt t getting the "right" xiom is to ask the following:

1.2. Generalization Step 1. What happens ifwe endow j itself, rather

than its factors, with the properties of preserving limits and colimits?Before attempting an answer, we should address the following philosophi

cal issue: Why should stronger versions of Lawvere's Theorem be true? Whenwe considered generalization, themotivation for concluding that certain largecardinals exist on the basis of generalizing properties of cowas Cantor's in

sight concerning the uniformity of the universe?no single cardinal shouldhave many special properties not found elsewhere in the universe. Here, a

principle that motivates us to conclude that "more preservation is better" is

Cantor's principle of maximum possibility: As much as possible exists. (See[15, pp. 20-23].) As we will see, when Lawvere's functor is endowed withever stronger preservation properties, the consequence is increasing combinatorial richness in the universe. This principle suggests to us that movingin the direction of strengthened preservation will lead to a more completerendering of the universe.

A theme that will develop as we move from Lawvere's result to refinementsis a strengthening of what we call here critical point dynamics. Histori

cally, perhaps the first nd most striking xample of this notion of criticalpoint dynamics occurs in the context of the canonical elementary embeddingj: V ?> VK/U = M, where M is the transitive collapse of the ultrapower

5This point will be developed further n the paper. What we have inmind here is that theadjoint factors of j have strong structure-preserving roperties (as inLawvere's Theorem),and that this situation "points to" the possibility of a stronger xiomatic formulation inwhich j itself as these or similar strong reservation properties.


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44 PAULCORAZZA

VK/U and U is a nonprincipal /^-complete ultrafilter over an uncountablecardinal n. The first point to observe is that the large cardinal strengthof such an embedding is located precisely at the first rdinal moved by the

embedding?namely, k?which must be a measurable cardinal. The secondpoint is that every set inM arises from dynamics that "live" in the vicinityof k. In particular, for every y e M, there is a function / with domain ksuch that y = j"(/)(?); here, the domain of / and the definition f itself,being defined from an ultrafilter over /s, epend on sets in the vicinity of n.InHamkins' terminology 16],k isa seed via j for U that generatesM.

In studying ritical oint dynamics from oth the set-theoretic nd categorytheoretic points of view, itwill be natural to define a critical point of a

mapping jto be the least cardinal y forwhich y V we consider here will never have the property that the leastordinal ymoved by is either non-cardinal or of cardinality |7*(y)|.)

An imprecise question that is nevertheless natural to ask as we considercritical point dynamics is,

"Why does V come equipped with a j: V ?> VI

What purpose does it serve?" If we think of set theory with large cardinalsas being full set theory, then one could consider this Lawvere functor, withwhich any ZFC universe isequipped, to be an intimation f a type f transformation hatwould have more fully eveloped properties n the presence ofthe spectrum f large ardinals; this fully eveloped version could be seen asthe evolutionary pinnacle of j. We have already suggested that embodiestransformational dynamics of K, since this iswhat functions do. Since an

obvious aspect of the transformational dynamics of V is the unfoldment ofall sets, we hypothesize that part of what j is "designed to do" is to generateall sets, in the way (or by analogous means) that a seed of an elementaryembedding j: V ?> M generates all sets inM. We formulate this hypothesisbelow and then, as a secondary theme as we seek strengthenings of Lawvere's

Theorem, we track the extent to which this hypothesis is verified in strongercontexts.

1.3. Critical Point Dynamics Hypothesis. It should be possible to generate every set in V from the auxiliary functor y, its critical point k, and sets

in the vicinity f k.

We will take a look at the critical point dynamics both for Lawvere'sfunctor and for each of the strengthenings we will consider. In each case, we

will see that the "type" of infinity hat merges from he embedding (whetherit is a large cardinal or, in the case of the Lawvere functor, simply a countablyinfinite et) does so at its critical point, and that, on the basis of dynamicsin the vicinity f this ritical point, "all" sets can be generated. We will callthis latter characteristic the seed property of the mapping.

Having at least roughly defined what we mean critical point dynamics, we

can now examine these dynamics in the case of the Lawvere functor. As we

observed earlier, in the presence of theAxiom of Infinity, hecritical point


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46 paul corazza

The idea behind the proof of (1) is that, given ameasurable cardinalifwe let D be a nonprincipal /^-complete ultrafilter on we can define afunctor j

= jD: V? K as follows: For any A", r and any A: Ar 7:

y (AT) XK/D = {[f] \ : B -> X and 5 GD}(f ~ g iff they gree on a set inD) (*)

j(h)([g])= [hog]The proof of the Trnkova-Blass Theorem shows that this j is exact, and

also that > whence j ^ id.For the converse, given an exact functor F: V

?V, not isomorphic to

the identity unctor, ne shows that the critical point nofF must be at leastas large as the least measurable cardinal (which must exist). Moreover, if khappens to be the least possible critical point among all critical points ofexact functors not isomorphic to the identity, then k must in fact equal theleast measurable cardinal. We prove this fact in Part 2.

For exact functors defined as in (*), it an be shown that the equivalenceclass [id]G jM of the identity unction d: k ?> as amember of nK /D, isaweakly universal element for : that s,for ny B and any ye j(B) there sf:K-^B such that./(/)([id])

= y. When D is a normal measure, as usual

[id] can be identified ith k. We will show that despite weak universalityof ft, t isnot the case here that every set in V isexpressible s j(/)(?).We pause to compare our formulation of a stronger Axiom of Infinity

based on a Trnkova-Blass functor

Trnkova-Blass Axiom of Infinity. There is an exact functor

j :V ?> V not isomorphic to the identity.with the formulation due to Lawvere, provably equivalent to the (original)Axiom of Infinity:

Lawvere Axiom of Infinity. There is a functor j: V -> V wherej: = G oF9 F H (?, and G is the forgetful unctor V? V.

Certainly, the Trnkova-Blass formulation is "cleaner" than Lawvere'ssince j itself rather than its factors) is endowed with the key preservation properties. Moreover, the price we pay for this cleaner formulation isnot great: we are simply requiring j to be exact, and exactness is a natural

geometric property of functors.

Nevertheless, we are seeking to strengthen the Lawvere formulation to the

fullest possible extent, and so we naturally seek further improvement. Oneaspect of the Trnkova-Blass formulation that could be further refined, at

least from the point of view of "critical point dynamics," is the fact that thecritical point of an exact functor not isomorphic to the identity is not always

guaranteed to be measurable (which is the large cardinal property associatedwith such functors), though the existence f the critical point k does at leastguarantee the existence of a measurable < k.


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48 paul corazza

j :V? V is an elementary embedding with critical point k given to us in a

model of the theory FC + BTEE. In [7] it is shown that kmust be weaklycompact, but not much more, because as we just observed, a transitive

model of ZFC + BTEE can be obtained from(less than)

0#.However,viewed as a functor, j is exact and not isomorphic to the identity. Therefore,

one might be tempted to conclude that, by the Trnkova-Blass Theorem,k ismeasurable. This conclusion does not follow, though, because in thiscase j is not definable in the universe V as itmust be for the Trnkova-BlassTheorem to apply. In particular, the Trnkova-Blass Theorem makes use ofan instance of Separation for j-formulas to define the ultrafilter D from j9and this instance isnot derivable from FC + BTEE.

To provide j with additional strength as we search for maximal preservation properties ofaj:V?>V, one further step we could take is to require jto satisfy all instances of Separation]. This additional requirement forces jto be very strong. Before investigating this "ultimate" theory, we first brieflysurvey the various strengths obtainable by adding certain natural instancesof Separationj. We begin with several definitions; the reader is asked to con

sult [19]and [20]for unfamiliar terms. A cardinal k is super trong f thereis an elementary embedding j: V ?? M such that n is the critical pointof j, M is an inner model, and Vj^

C M. A cardinal k is extendible if,

for every ordinal rj > k, there is an ordinal ? and an elementary embeddingi: Vrj > V{ such that crit(z)

= k and rj i(k) < ?; themap / is called anextendible embedding with critical point k. For each n G co, k is n-huge ifthere exists an inner model M and an elementary embedding j: V ?> M

such that crit(y') n andM isclosed under ./"(/^-sequences; '{k) iscalledthe target of j and j is called an n-huge embedding, n is super-n-huge if,for every X > k, there is an fl-huge embedding j such that crit(y') = kand > X. The axiom \^{k) asserts there is an elementary embedding

j:Vx

?>Vx

with criticalpoint

k and X> k alimit;

n this casej

is calledan /3 embedding and k, an cardinal. The axiom I2M asserts there is an

elementary embedding j: V ?> M having critical point k so that the inner

model M includes as a subset Vx where X is the supremum of the critical

sequence of j. The axiom Ii(k) asserts there is an elementary embedding

j: Vx+\ ?> Vx+\where X is limit nd k < Xis the critical point; in this ase,j isan I\ embedding nd k isan I\ cardinal. The axiom In(?) asserts there san elementary mbeddingL( V^\) -? L( Vx+\) having critical point k, whereX> k isa limit; n this ase j isan Iq embedding nd k isan lo cardinal.

Measurable Ultrafilter Axiom of Infinity. The class {X c k:

k G ](X)} isa set.

Adding theMeasurable Ultrafilter Axiom to ZFC + BTEE is the resultof adding the single instance of Separation] that asserts that the ultrafilterover k defined by j is a set. Perhaps surprisingly, for a theory to assert in


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50 PAULCORAZZA

Each of the extensions of ZFC + BTEE just considered here asserts theexistence of an elementary embedding j: V ?> V with certain additional

properties. Any of these could be considered candidates for a foundational

extension of ZFC. Still, our aim has been to strengthen the preservationproperties of the Lawvere functor given at the beginning to the furthestpossible extent. Our way of doing this is to introduce as additional axioms to ZFC + BTEE all instances of Separation for j-formulas denotedSeparation]). The theory BTEE + Separation] is known as theWholenessAxiom orWA. A model of the theory FC + WA provides us with a verystrong structure-preserving map of V to itself. We first establish that, relative to some of the very strongest large cardinals known, ZFC + WA is notinconsistent:

1.10. Consistency Theorem [4]. If there is an 1$ embedding i: Vx ?> Vx,then there s transitive odel ofZFC + WA.

In this case, the model is (Vx, G, /). The following theorem summarizessome of the main results concerning the theory ZFC -I-WA; outlines of

proofs will be given inPart 2. We begin with the followingmeta-definition(which, as we discuss below, may be expressed formally in the language of

ZFC + WA): A cardinal k is aWA-cardinal if there is aWA-embeddingj: V ?> V having critical point k.

1.11. The Wholeness Axiom Theorem [4,7]. The following are characteristicsof the theory FC + WA:

(1) Assume ZFC + WA and that : V? V is theWA-embedding, with

critical point k. Then k is super-n-huge for every n; moreover, there is a

proper class of cardinals that are super-n-huge for every n.

(2) ZFC + WA is indestructible nder etforcing.(3) The only "natural" inner odel ofZFC +WA, if there sone at all, isV

itself.(4) The critical equence (k, j(j(?)),...) forms ay classof indiscernibles

for V. That is, or any e-formula


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V ?*> V 51

are those that behave in the expected ways. In Part 2 of the paper, we givedefinitions nd motivation for this concept. Part (3) of the Theorem ofcourse depends on our definition of "natural model"; if the reader finds

the intuitive otivation for this definition ompelling, then part (3) of thetheorem isquite striking: t suggests that a "good choice" for amodel ofZFC + WA is V itself; hat, in an unexpected way, V is even the canonicalmodel, being the only reasonable inner model of the theory.

Part (4)of the theorem highlights the sense of "ultimacy" that ccompanies a large cardinal axiom such asWA. It tells us for example (speakingsomewhat loosely) that every large cardinal property that holds true of k

must also hold of every in the critical sequence of j;moreover, sincethe critical sequence isunbounded inON (seePart 2 formore on this oint),every large cardinal property true of k holds true for unboundedly manycardinals in the universe.

A slightly tronger actof this ind is mentioned in art (5)of the theorem:The large cardinal property of k that isderived from theWA embedding jalso holds for each member of the critical sequence. We call this the selfreplicating roperty fWA cardinals. This property an be formulated ithinthe language {g, j} as follows:Given thatWA holds with embedding j andcritical point k, then every kn in the critical sequence of j is aWA-cardinal,

where ? k and kn = The idea here is that the usual iterates byapplication6 j, = j j, = j (j j),... are definable in FC +WA fromand, ingeneral (V, g, j[/1+1])samodel of ZFC + WA that witnesses the factthat kn is aWA cardinal. Thus the metatheorem can be stated formallyas a ZFC + WA schema as follows: For each e-formula (x\, 2,..., xm)9ZFC + WA proves each of the following:

V? g CQVX\,X2,...9Xm [(x\,X2,...,Xm)

These consequences together with the fact that ZFC + WA h \fn gco(crit(j[w+11) kn) state in a formal way that each term of the criticalsequence for j is aWA-cardinal.

It could be argued that dding Separation^ toZFC+BTEE is bit artificial,even though the consequences are quite nice. An alternative to Separation^

6Application j-j isdenned in the usual way: For any j-classC, define j C by

j-c= (J j(cnaaeON

InZFC + WA, such definitions make sense sinceC n Va isa set. It is easy to see that

j-j= U id raEON


17/49

52 paul corazza

which was infact our first hoice inour earliest ttempts 8]at studying heseaxiom systems, is the following:

Amenabililtyj Axiom of Infinity. For every set x, the restriction

j \ isalso a set.In the literature, he theory BTEE + Amenability is denoted WA0; notethat Amenability- is provable from Separation]. It is shown in [7] thatZFC +WAo has the same strong arge ardinal consequences asZFC +WA;that is, part (1) of the Wholeness Axiom Theorem still holds. However,

ZFC + WAo is not known to be indestructible nder set forcing. Thislimitation eads us to believe that FC + WA is the better choice.

As for the ritical point dynamics of an embedding that omes from A,part (1) shows that the critical point has nearly the strongest ossible largecardinal properties. Such a,j:V?>V has a different sort of seed property.

We will show inPart 2 of thepaper that t cannot be the case that very set isexpressible as j for / e V. However, j has amuch stronger propertythan is found in the other examples we have considered. We state the resulthere and outline the proof in Part 2:

1.12. WA Critical Point Theorem [4]. Let j: V -> V be a WA

embedding with critical point k. Then there is a function /:??? VK such

that for every set x, there is an elementary embedding i: ?> satisfying

(1) k is the ritical oint of i\(2) /(/)(?)= x.

Indeed, we may write

V = {/(/)(?) | / san extendible mbeddingwith critical oint k}.

The idea behind theWA Critical Point Theorem is that, in the spirit ofLaver sequences for supercompact cardinals [24], one can show [3], [4] that

if there is an extendible cardinal k, there is an "extendible" Laver sequence

f'.tv ?> VK (definitions and details are given in Part 2). As the referee pointsout, theWA Critical Point Theorem, as given here, could be restated more

simply y replacing rjwith k + 1 and ( with i(rt) 1.We will give a slightlyenhanced version of this theorem nPart 2 forwhich this simplification illnot be possible.

Applying the xiomatic framework FC + WA to the test uestion. In thissubsection we review some of the work done in [3], [4], [5], and [17] to

address the Test Question ("Which large cardinals admit their wn brandof Laver sequence?") in the context of the theory FC 4-WA. We offerthis as an example to illustrate how a foundational axiom for large cardinals

may contribute to the general program of large cardinal research. Here,we indicate how the use of WA as part of the axiomatic framework not

only ensures the existence of themany different inds of large ardinals thatnaturally arise, but also becomes an integral part of the solution and even


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V ?* V 53

suggests direction for further efinements f the solution inwhich WA isreplaced with weaker axioms.We begin with a convenient uniform definition of a broad class of large

cardinals that are candidates for admitting some type of Laver sequence. Theintuition is that the large cardinals belonging to our class are those that ariseas the critical point of some elementary mbedding of the form Vp

? MwhereM isa transitive et.Our background theory FC + WA ensures theexistence of many such classes; as it happens, it also guides our research intothe question. Here is a more precise definition of the classes of embeddingsthat concern us:

Let 9(x, y, z,w) be a first-order ormula in the anguage {e}) with all freevariables displayed.We call 9 a suitable ormula if the following entence isprovable inZFC:

Vx, y, z, w [9(x, y,z,w) => "w is a transitive set" A "z is an ordinal"A "x: Vz ?> w is an elementary embedding with critical

point y"].

For each cardinal k and each suitable 9{x, y,z,w), let

?eK={(i,M): 3pO(i,K,p,M)}.

Intuitively, QKonsists of a class, defined by 9, of elementary mbeddingsVp

? M; the need to pair up embeddings with their codomains in thedefinition f ?eK sa technicality hat is explained in [4].

We show in [4]how many of the familiar globally defined large cardinalnotions, such as strong, supercompact, extendible, superhuge and superalmost-huge cardinals, can be characterized as classes of embeddings of this

kind. Of course, having ZFC + WA as the background theory ensures theseand plenty of other such classes exist in the universe.

If such a class is to have any hope of admitting some kind of Laversequence, it needs to have the additional property of regularity, which assertsthat for any set x, there is some i: Vp

?> M in the class ? for which x e M.More precisely:

1.13.Definition (RegularClasses). A class ?6K sregular f

\ty>n^P>y 3i 3M [9(i,k, /S, ) A /(/c) yA Vy CM ].

It is shown in [4] that the classes corresponding to each of the five globallarge cardinal notions mentioned above are in fact regular classes. We cannow define a notion of "Laver sequence" for any class ?eK\ it can be shownthat if such a class does admit such a Laver sequence, itmust in fact be aregular class.


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54 PAULCORAZZA

1.14. Definition (Laver sequences). Suppose is a class of embeddings,where 0 is a suitable formula. A function g: k

?VK is said to be ?%-Laver

at k if for each setx and each X> k there re ft X,and /: Vp M e ?eKsuch that /(?) > Xand /(g)(k) = x.We show in [4] that our definition coincides with the usual definitions

of Laver sequences; for instance, if 9 is a formula that defines a class ?eKequivalent to supercompactness of k, then /: k ?> VK is Laver in the usualsense if nd only if t is ?%-Laver.

Now that we have a definition of Laver sequence, we can ask, in the spiritof the Test Question, "Which classes ??K admit Laver sequences?"

In [4]we observe that n application of theWA-embedding j gives a hint

about building Laver sequences generally: Let U be the normal ultrafilterover k derived from : U = {X c k |k g j(Ar)}. If/: /c > VK isa Laversequence (in the usual sense), it follows that

{a < k \ \ : a ?> Ka isLaver at a} g 17.

From this observation, one could attempt to build /: n ?? FK generally byarranging it so/ (a) isarbitrary f \ isLaver at a (and the intention sthat this ondition holds for most" cardinals a), whereas f{a) isawitnessto the failure of Laverness at a otherwise. This idea can be used to obtain theusual Laver sequences for supercompact and strong cardinals without theuse ofWA. However, in the presence of the WA-embedding, one shows theconstruction yields Laver sequences for virtually any globally defined largecardinal notion; for instance: superhuge, super-almost-huge, and extendiblecardinals. In fact, we show [4] that whenever class ??K is regular and is

sufficiently compatible" with the WA embedding, itmust admit a Laversequence. Here we observe the helpful role played by the embedding j thatis now part of the background axiomatic framework. For completeness, we

give a more precise definition of "compatibility."1.15.Definition (Compatibility). ?dKis said to be compatible with the

WA-embedding j if for each X, k < X < j(k), there exist f},i such thatX < ft < /: Vp -> M, (/, f) g ?BK, nd i is compatible with j \Vp: Vp ?> N

=Kj(^) in the sense that there is k: Af

? N such thatk o i= j rVp and k \Vxn M

= id^nM

Vp-^ vm

\ kAfTo complete the icture, we show in [3]from FC+WA that not all regular

classes necessarily admit a Laver sequence. Here is a summary of results:


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V ?> V 55

1.16.Generalized Laver Theorem [3,4]. (ZFC+ WA).(1) If?QKs regular lassof emebeddings ompatiblewith the A-embedding,

then there is an ?eK-Laver sequence.

(2) There isa transitive odel of FC inwhich there sa regular lass?eK fembeddings that does not admit an ?%-Laver sequence.

Our purpose in reviewing these results is to illustrate the role of an axiomatic framework for large cardinals. In this example, the theory ZFC+WA

guarantees the existence of the many classes of large cardinals that could admit Laver sequences. It then also plays a direct and natural role in theconstruction of a general notion of Laver sequence as well as providing acriterion to decide which classes admit Laver sequences. A third contri

bution ofWA in this ontext is that t points theway for further efinementsinwhich results of this kind are obtained under weaker hypotheses than WA.

For instance,we obtained in [4]similar results by replacing with a class ofembeddings of the form i: V ?> N, modulo several technical conditions.The point is that the xiomatic framework FC + WA not only provides inthis case an "ontology" of large cardinals in the background but also plays adirect role in formulating a solution to the question and even serves to guidefurther research beyond the direct use ofWA.7

The naturalness of WA as a foundational axiom. Even ifwe think it is reasonable to admit into the formal universe all the known large cardinals thatare used today in research, the question about how to state this acceptanceaxiomatically would remain. An axiom that simply asserts the existence ofcertain large cardinals lacks the cogency we might expect to find in a foundational axiom. In addition, as we have pointed out, an optimal extension of

ZFC would have other desirable properties, such as indestructibility underset forcing. It would also be more in the spirit of foundations to introducean axiom that is

clearlymotivated

byour

knowledgeof the structure of

the universe on the basis of ZFC. Certainly the early simple heuristics hadthis characteristic, using basic principles obtained from Cantor's early vision

(and those of others) of theuniverse as guidelines for enhancing the richnessof the universe via large cardinals.

Taking into ccount issues f this ind, the olution provided byZFC+WAhas many advantages. As we have seen, it is strong enough to derive vir

tually all the large cardinals that arise in research and it is indestructible

7Evenwithout a formal theory f large cardinals, the strongest arge cardinal notions havealready been used in a similar way. For instance, as we mentioned earlier, the first proofof determinacy of sets was obtained byMartin assuming an h embedding, and the firstproof of projective determinacy was obtained byWoodin using an lo embedding. Many ofthe insights n these proofs were refined to obtain the optimal large cardinal hypotheses forthese results that are known today. The difference s that the strong large cardinal axiomsused to obtain the arly results had tobe invoked s ad hoc and somewhat amazing hypothesesrather than as part of a formal axiomatic system.


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56 PAULCORAZZA

by set forcing. It also bears the character of the "ultimate" large cardinalaxiom because of the self-replicating feature that if there is aWA cardinalthere must be unboundedly many such cardinals above k. Moreover, it

has long been recognized [22], [28] that an axiom of the form "there is anontrivial elementary embedding j: V ?> V" represents a kind of ultimatelarge cardinal axiom in another sense: The strongest large cardinals are defined using elementary mbeddings of the form : V ?>M; and one of thecharacteristics f the stronger arge cardinals is that the imagemodel M ofthe corresponding embedding(s) tends to includemore of V. For instance,a superstrong mbedding requires V^ C M and a huge embedding re

quires j^M C M. Therefore, in a sense already familiar to large cardinal

researchers, some strong but not inconsistent version of an elementary embedding j: V ?> V would be the natural culmination of all large cardinalaxioms.

We consider "naturalness" ofWA from another point of view in this paper.Recall thatWA is the last of a series of refinements f a certain form f theAxiom of Infinity?an axiom that we already take to be incontrovertible.That first form of the axiom, obtained from Lawvere's Theorem, alreadytells us that once we accept the actual infinite into our universe, the universe becomes

equippedwith a certain

map j:V ?> V that exhibits

strongpreservation properties. Our analysis began with this natural and incontrovertible starting point and proceeded by asking, "How can this axiombe optimized?" Moreover, refinements of the axiom always proceeded inthe direction of increasing combinatorial richness of the universe, adheringfairly closely to Cantor's principle of maximum possibility. Therefore, sincethe form of the axiom WA, asserting the existence of a certain j: V ?> V,

originates with an axiom of ZFC, and since the theme of refinement s anatural one, we suggest that WA meets the requirement of "naturalness" of

a new foundational axiom.Despite our belief that FC + WA isa strong andidate for an extension

of ZFC that accounts for large cardinals, one can reasonably argue thatthere is still room for improvement, and also that there are other acceptablealternatives; we consider these points in the next section.

Limitations of the theory ZFC + WA and an alternative axiom schema.

The following sa list of legitimate oncerns about the theory FC + WA.We make some remarks about each point, and then discuss an alternative

toWA, introduced yWoodin, which offers mprovements n some of theseareas.

1.17. Limitations of ZFC + WA.

(1) From ZFC + WA, cardinals that are super-w-huge for every n are deriv

able, but none of the cardinals given by the axioms I3-I0 are derivable,

though, as was discussed earlier, some of these have played a significant


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j: V V 57

role in the early proofs of core theorems concerning properties of setsof reals.

(2) ZFC +WA is formulated n the expanded language {e, j}.This dimin

ishes its intuitive appeal.(3) In a ZFC + WA universe (KG, j), the critical sequence of j is cofinal

in the ordinals. This again detracts from its intuitive ppeal since itviolates our intuition that the universe is "inaccessible"?an intuitionthat s formalized by theReplacement Axiom.

Points (l)-(3) seem to the author to be legitimate oncerns; we make afirst attempt to address them here, though we do not claim to have entirelyresolved these issues yet.

For (1), one could invokeCantor's reflection rinciple tomotivate theexistence of an I3cardinal in the universe, ssuming ZFC + WA holds true;even this approach, however, does not account for the axioms I2?lo

For (2), a form of this problem has been present as long as the question"Is there a nontrivial elementary embedding j: V ?> VT has been around.Since it snot possible to formalize the uestion in the anguage ofZFC, evenwith the use of proper classes (without uantification), it is inevitable thatsome extended framework must be used to study this question. In Kunen's

theorem [22], inwhich he proved that no such embedding could exist, heworked in a class theory. Doing so forces us to regard j as a class, whichthen must obey the instances of Separation and Replacement; this additionalaxiomatic baggage plays a crucial role in the proof of inconsistency. Toexamine the actual strength of elementarity therefore requires a differentcontext for studying the question. The only way we know to do this isto expand the language in the way that we have. Therefore, though the

language {e, j}may not be appealing, itmay be unavoidable ifwe wish tostudy elementary embeddings V ?> V without the impact of class

theory.To address point (3), we suggest an alternative perspective, arguing thatthe spirit ehind Replacement isnot truly eing violated in FC +WA afterall. The original intuition about Replacement was that, because the universeis vast and, in Cantor's words (cf. [15, p. 44]), "beyond mathematicaldetermination," in forming a countable (or longer) sequence of sets, theresulting sequence should not be cofinal in the universe. This intuitioncontinues to be realized in a ZFC+WA universe relative to e-formulas, since

Replacement for such formulas continues to hold true inZFC+WA. Indeed,

the only sort of violation of this intuition that could occur isvia j-classes.And, of course, the sets that arise in "ordinary" mathematical practice wouldnever require the use of j. Because j isa very different ypeof entity, hichencodes (via an extendible Laver sequence) all sets in the universe, itmaynot be too heretical to consider that V could appear different rom 'spointof view, and that this point of view is not accessible to inhabitants f V.

Analogous situations in set theory are commonly encountered. For instance,


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58 PAULCORAZZA

consider L's view of the F-cardinal cow, assuming 0# exists. Let d = {cow)vSince 0# exists, 3 is a Silver indiscernible, ence inaccessible inL. This

means that from "inside" the world of L, Vs appears to be a universe for

mathematics, a transitive model of ZFC. The knowledge that there is acofinal co-sequence f:co-> is not available to L inhabitants. This co

sequence is an altogether different ypeof entity relative to V\\ though itis a subcollection of V^, it is neither a set nor a (definable) class relativeto this universe. What grants / its special status is the existence of 0#?

something that is beyond the comprehension of L. In a similar way, ina ZFC + WA universe, what grants the co-sequence k,,)(k), j2(/s),... its

special status in V (the status of being a cofinal co-sequence that is not

G-definable in V) is the presence of j?transformational dynamics thatare

outside the comprehension of V. Therefore, any non-Replacement-likephenomena wemight find in ZFC +WA universe remain forever nknownto the inhabitants f V. For the business of themathematics of sets andG-classes, no violation of Replacement can ever occur in a ZFC + WAuniverse.

To conclude this section, we give a brief discussion of a very strong axiomschema due toWoodin which could be used inplace ofWA.8.

Weak Reinhardt Axiom (WRA). There is a j: VM?

VMwith critical point k such that VK


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j: V V 59

Proof. Let k be a weak Reinhardt cardinal. For the first part, let (x)bethe following formula:

(x) "x is an ordinal" A 3y> x 3d > y3e: Vs+\?

K^+i (crit(e) = y).First we claim that VK \=Va 0(a): Fix a < /s. Since k is an Ii cardinal,V |=0(a). Since VK < V, VK \= (a). Since a < was arbitrary, heclaim follows. But now, from the claim and the fact that VK< V again, itfollows that V |=Va 0(a), and we are done.

For the second part, let /: Vx?> Vxbe the restriction f an Ii embeddingto Vx,where X is a limit nd crit(i') = k. One can argue as in part (1) oftheWholeness Axiom Theorem (given npart 2 of this paper), insideV^9toshow that the statement "there are arbitrarily large cardinals that are super

s-huge for every n" holds in Vx. Since VK - Va such that

V = {/(/)(a) | / s an extendible embedding with critical point a}.One disadvantage of the strong large cardinal consequences of WRA is

that it is difficult o assess whether WRA isconsistent; unlikeWA, it is notknown to be consistent relative oany of theusual very strong arge ardinalaxioms.

Theonly

well-knownlarge

ardinal axiom that s not known tobe derivablefrom WRA is lo- Thus, even this seemingly all-encompassing axiom is not

quite able to account for all the known large cardinals.One could argue that the language inwhichWRA is formulated smore

natural than {e, j}, but note thatWRA isnot formalizable inZFC: If itwere, let k be the least weakly Reinhardt cardinal; then VK?and therefore Vitself?satisfies the sentence "there is no weakly Reinhardt cardinal", whichis impossible.WRA can be formulated n the language {?, c},where c isanew constant symbol ntended odenote the ritical point k of an embeddingj: Vx+\ ? Vx+\. An infinite ollection of axioms could be used to assertformally that VK -


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60 PAULCORAZZA

critical point n. Therefore, Vs isa transitive odel of ZFC +WR(?). (Thisargument can be formalized using two constant symbols with appropriateaxioms.) Therefore, the existence of twoweak Reinhardt cardinals impliesthe existence of a transitive model of one weak Reinhardt cardinal. Ina similar vein, from model of ZFC together with arbitrarily argeweak

Reinhardt cardinals one can obtain a model of ZFC inwhich the weakReinhardt cardinals are bounded in the universe: Starting with a model(M9E) of arbitrarily argeweak Reinhardt cardinals?formalized in someexpansion of the language of set theory?one can obtain amodel (N, E,k,S)(in the language {G,ci,C2}) inwhich k and S are both weak Reinhardtcardinals, k < S9and for all ywith k < y < S, y isnot weak Reinhardt.

Then,in

N9 Vssatisfies FC+WRA and also "theweak Reinhardt cardinals

are bounded in the universe."The conclusion that we draw from these bits of reasoning is that the

existence of a weak Reinhardt cardinal does not imply the existence of othersabove it.This fact leadsus to conjecture that FC +WRA can be destroyedby set forcing?moreover we conjecture that collapsing k to a countableordinal would be enough to destroy WRA. Of course, as Proposition 1.18

shows, no such forcing can alter the fact that Ii cardinals and cardinals thatare super-w-huge for every n must pervade the universe.

The reasoning ust given also shows that weak Reinhardt cardinal typically fails to havemost of the strong, lobally defined large cardinal properties, ike xtendibility nd superhugeness. Suppose for xample that k isbothextendible nd weak Reinhardt with Ii embedding : Vx+\

?V^+i. Lete) be

an inaccessible bove Xwith VK< Vs (this follows from xtendibility). owVs is a transitive model ofWRA. Thus, WR(?) + "/c is extendible" is strictlystronger than WR(?). (We do not know if the same can be said for super

compactness. It does follow from WR(/c) that k is strongly compact since

itmust be a measurable limit of supercompacts.) Therefore, as mentionedabove, it does not follow from WR(/c) that there is an /: n ?> VK such that

V = {i(f)M | i isan extendible embedding with critical point k} since theexistence of such an / implies n is extendible.

Perhaps the biggest drawback toWRA in our view is that its motivation(as far as we know) ispurely technical: It arises from taking the strongestform of an elementary mbedding from rank to a rank not known to beinconsistent, nd ensuring a global effect y requiring VK -< V. It is not

clear from"first

principles" whyan axiom such asWRA should hold true

in the universe. And sinceWRA isnot known to be consistent relative toother very strong (andwell-known) large cardinal axioms, it isdifficult obe entirely confident about it as a candidate for a new foundational axiom.

We conclude this section with a table comparing relative strengths and

weaknesses ofWA and WRA.


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :v -* v 61

Property of Axiom Schema WA_WRA_Consistency Consistent Not known to be

relative to an I3 consistent

embedding. relative to other

strong, wellknown axioms.

Naturalness Motivated by Primarily technistrengthening cally motivated:

preservation Weaken inconsis

properties of a tent j: Vx+2 ?>

map j: V? V V^+i by lowering

that is provably X 2 to X 1 ndequivalent to then globalize bytheAxiom of requiring VK - V is anatural culmination of embed

dings j: V??

Mthat define moststrong largecardinal notions.

Large cardinal consequences Critical point k There is a

issuper-H-hugefor proper class of

every n, and there Ii cardinals andis a proper class of cardinals thatsuch cardinals. are super-?-huge

for every n.

However, thoughcritical pointk is stronglycompact, it lacks

many other

global largecardinal prop

erties, such asextendibility and

_superhugeness.Robustness under set forcing WA is indestruc- WRA isnot

tible under known to beset forcing. indestructible un

_ der set forcing.


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62 PAULCORAZZA

"Inaccessibility" of the I nZFC +WA, the I ZFC + WRAuniverse length-co critical universe is inac

sequence ofj

is cessible, withoutcofinal in ON. qualification.

Replacement continues to hold fore-formulas,

so the universe for

"ordinarymathematics"continues to be

inaccessible.Language inwhich theory s {?, j} {? >c)formalized

"Ultimacy" of the axiom WA cardinals WRA impliesschema have the self- there is a proper

replicating prop- class of Ii cardi

erty: If k is a nals and ofWA cardinal, un- cardinals super

boundedly many w-huge for everycardinals above n. But the exisk are also WA tence of one WRcardinals. cardinal does not

imply xistence ofothers above it.

Critical point dynamics Critical point k There is (differgenerates all sets. ent from critical

point)that gener

ates all sets.

?2. Selected proofs of the main results.

Preliminaries. In this section we give a concise review of category-theoreticconcepts that are used in the paper. We assume the reader knows the definition of a category and the assortment of standard constructions thatare done in categories: (finite) products, (finite) coproducts, equalizers,coequalizers, terminal objects, initial objects, and exponentials. We alsoassume the reader knows the definitions of monic, epic and iso arrows, andof functor and natural transformation. See [26] as necessary.

For any category C and objects a, b in C, the setHome (a,b) is the set ofall arrows a ?> b in C.

A diagram D in a category C is a collection of C objects {df \ El}together with some C arrows d\ ?> dj between some (or all) pairs of the


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V ?+ V 63

objects inD. A cone for a diagram D consists of a C-object c together witha C-arrow /,-: c

?di for each object d\ inD such that the diagram

di->- dj

c

commutes, whenever k is an arrow in the diagram D. The notation {/, :c ?> di} signifies a cone for D. The dual concept is a co-cone {gi: ?> c},consisting of an object c and arrows g/: di ?> c for each di in Z>.

A /i/w# or a diagram D is a Z)-cone {/, : c ?> rf,-} with the property that

for any other D-cone {/* : c'?>


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64 PAULCORAZZA

Suppose F: C ?> V is a functor and c e C,d e V. A X>-arrow u: d ?>

F(c) isa universal rrow iffor ny x G C and any g: d F(x) inV, thereisa unique /: c ?> x inC such that g = F(/) o w.

c C are functors. Then F is left djointto G, and we write F H G, if there is, for each c G C,d GD, a bijection

^: Homp(F(c),rf) ?> Homc(c, G(d)) that isnatural in c and In this

case (F,G99) is said to be an adjunction. For each c G C, let //c enote#c r(c)(l/r(c)); that is, c is the imageunder # of the identity rrow living nHomx>(F(c), F(c)). One shows that rjc: c ?> G(F(c)) isa universal arrowfor each c and that the collection {rjc \ G C} are the components of anatural tranformation 77: \q ?> G oF. 77 s called the unit of the adjunction.

An adjunction is completely determined by its unit. That is, given functorsF: C -+V and G: V ?> C and a natural transformation rj: \q ?> G oF such

that, for each c G C, ^ : c ?> G(F(c)) is a universal arrow, then FHG.Whenever F H

G,F

preservesall colimits of C and G

preservesall limits

of V.The category of sets, denoted Set, has as objects all sets and as arrows all

functions between sets. For any category C, the category of endos from C,denoted C^, has as objects all C-arrows c ?> c. Given /: c ?> c,g: d

?

G an arrow a: / ?> g is a C-arrow ea\ c d that makes the followingdiagram commute:

/c-^ c

d-^d

Suppose F: C ?> Set is a functor nd c G C. An object u G F(c) is a

weakly universal lement or F if for each J G C and each y e F(d) there san : c ? d inC so that (fd)(u) = j>;more verbosely, is said to beweakly represented y c with weakly universal lement u. Moreover, if ^ isunique for each choice of d, then u is a universal element for F; again, one

also says in this case that F is represented by c with universal element u.From a foundational point of view, one siginficant eature f a (weakly)

universal element u for a functor F is that it provides a way of reaching a

vast expanse of sets from a single "seed" u. For our purposes, it will be

useful to know whether every et in Set can be reached in thisway. We willdeclare that a functor : C -> Set is cofinal if for every x G Set there isc G C such that x GF(c). One easily verifies hat ifu isaweakly universal


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V ?> V 65

element for functor : C ?> Set, then every set is expressible as F(f){u)for ome arrow / inC if nd only if iscofinal. (From the ategory-theoreticpoint of view, this definition of "cofinal" is rather unnatural (as the referee

has pointed out) because it is not preserved by natural transformations.This notion and its (set-theoretic) connection to a universal element hasturned out to be conceptually helpful, so we have used it advisedly; see

Theorem 2.17.)Any model (M,E) ofZFC

? Infinity an be turned nto cartesian closedcategory M as follows: The objects are the elements ofM. Given a, b e M,

fa ?> b is an arrow in the category if and only ifM f= "/: a ?> b is afunction". Since, internal toM, the usual set-theoretic product a x b and

exponentiation ab operationscan

be carried out, M is cartesian closed. Forconvenience,we will denote this categoryM instead ofM. We will refer oa model of this kind as a category of sets.

Lawvere's Theorem. As we mentioned in Part 1,Lawvere's Theorem is infact a theorem about cartesian closed categories. For this paper, though,

we restrict our attention to categories obtained from models of ZFC?

Infinity, s described in the section on Preliminaries. We fix amodel V ofZFC ? Infinity, which we treat as a category of sets. Recall that the theoryZFC ?

Infinity -Infinitysequivalent

tofirst-order eano Arithmetic(PA)in the sense that a model of one can be interpreted in a model of the other

(see [10]for discussion); as inPA, wemay invoke theprinciple of inductionand also define classes via recursion.

As a preliminary to the proof of Lawvere's Theorem, we need to introducea category-theoretic construction for the set of natural numbers. Recallthat if category admits a terminal bject, then ll terminal bjects in thecategory are isomorphic and that, in a category of sets, any singleton is aterminal object. Terminal objects are typically denoted with the numeral 1.

A natural numbers bject (NNO) ina category C isa triple X,z, s) where zand .s*re arrows forming a diagram

1 X X,

with the following niversal property: For any diagram 1 U -UU inC,there s a unique arrowX ^ U such that the following diagram commutes:

X---^X

1 h h

u-'?^ u


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66 PAULCORAZZA

Clearly, the definition f anNNO at leastmakes sense in ny category thathas a terminal object, though not every such category actually has such an

object. If each of (U,w, ) and (W,w, t) isanNNO, there s an isomorphismbetween them; indeed, the unique h :U

?W guaranteed by the universalityof NNOs is an iso in the ambient category.

We outline briefly hy the assumption that theAxiom of Infinity oldsin V implies there is an NNO in V. The Axiom of Infinity mplies coexists, and we can define the usual successor function succ: co ?> co : n i?>n U {?}, as well as the map z: 1 ?> co : 0 i-? 0. Now, the assertionthat (co, , succ) is an NNO is equivalent to definition by primitive recursion:

succCO -> CO

1 h h (1)

u--?^ u

Thediagram

can be seen asdefining (unique) sequence

h: co > U fromt: U ?> U by primitive recursion:

Mo) = 7(0),h(n \) t(h(n)).

The commutativity f the triangle n the diagram (hoz = y) isa statement fthe base case of the recursion; the commutativity of the square in the diagram(h o succ = t o h) is a statement of the induction step of the recursion. Since

definition by recursion is provable from the existence of co, it follows thatone can obtain anNNO (namely, co, , succ)) from theAxiom of Infinity,working inZFC

? Infinity.To prove the converse inZFC - Infinity?that the existence of anNNO

implies the Axiom of Infinity?start with an NNO (X,z,s). We showthat X must be infinite. Define by recursion the class sequence F =

(X0,Xi,*2,..., *?,...) by

x0=

z(0)=

s?(xo),xn+i = s(xn) = sn+l(xo).

By Separation, the range of F is a set (being a subclass of the setX). Onenow uses induction nd the universal property ofNNOs to show that, foreach n, xo>*i,*2> ,xn are distinct. (For the induction, formally, one

uses the formula that defines F to define the induction P(x): P(n) asserts


32/49


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68 PAULCORAZZA

\A XsuccA x co-> A x co

A h h

?-?JiWe do not give the proof here. The main idea rests on the connection to

the concept of definition by recursion. In the diagram, ifwe letB = A,define xo: A ? A x coby xo(a) = (a, 0), and define /: A ?> A = 1^, thenthe

uniqueh: A x co > A that s

given bythe theorem s the

uniqueh

givenby the efinition ByRecursion Theorem for the data

h(x, 0) = x,

h(x,n + 1)= g(h(x,n)),

where x g A and g: ^4?> A are given.We proceed now to the proof of Lawvere's Theorem. For the reader's

convenience, we restate the theorem:

Lawvere's Theorem. Suppose V is a model oj ZFC - Infinity. Then thefollowing are equivalent:

(1) V satisfies theAxiom of nfinity.(2) There isafunctor j: V ?> V that actors as a composition G oF of

functors satisfying:(A) F HG (F is left djoint toG),(B) F :V -> V?9(C) G: V? -> K wrte forgetful unctor, efined byG(A ^ A) = A.In particular, F preserves all colimits and G preserves all limits.

V-^ V

V0

Proof of Lawvere's Theorem (Outline). As before,we fix model V ofZFC ? Infinity. Let G: ?> V denote the forgetful functor. If /: a ?> a9g: b ?> b are objects in V?, and a :f ?> g is an arrow in V?, note that

G(f) = a9G(g) = b, and G(a) = ea: a ?> fe. Suppose F isa left djointto G, defined in V. Let j = G oF be the Lawvere functor. We exhibitarrows z, s such that (y 1), z, s) is anNNO, fromwhich it follows that theAxiom of Infinity olds (by the NO Lemma). Moreover, we show that thecritical point for is 1.


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :V ?* V 69

First we observe that y(0) = 0: Being a left djoint, F preserves allcolimits, nd so, inparticular, preserves the initial bject 0 of V. It is easyto verify hat the initial object of V? is the emptymap 0 ?> 0. It follows,

therefore, hat y(0)=

G(F(Q))=

G(0-+

0)=

0.Next, we apply to 1 and show that (1) is theunderlying et of anNNOand, inparticular, that |y(l)| > 1.F(l) isan object s: X ?> X in V? andG(F(1)) = X. Let rj: 1y ?> G oF be the unit of the adjunction of F and(?, and let z = rj\: 1?? (/(.F(l)) be its component at the object 1.We usethe fact that z is a universal arrow to show that

1l^IisanNNO:

Supposewe are

given F-maps1 {/ (7. Note that s and

g are objects in V?. By universality f z, there s a unique a :s g suchthat j>= G(a) o z = ea o z.

J 1-^GF(l)G(a)

In other words, ea isunique such that the following iagram commutes:

X-^X

1 ea ea

u-?>u

In particular, (X, z, s) isanNNO, as required. As j(0) = 0 and y (1)= X,which is infinite, e have also established that 1 is the critical point for .

We now prove the other direction of the theorem. Assuming the truth ofthe xiom of Infinity, ur proof of the NO Lemma givesus that co, , succ)is an NNO, where x: 1?> co: 0 i?>0 and succ is the usual successor function.

We define left djoint F: V ?> V? for the forgetful unctor : V?-> V asfollows: For any setA, F(A) isdefined to be the endo Axco 1-4XSUCC)xco.

Given a F-arrow /: A ?> B, let (f) be the F?-arrow / x lw: l^x succ??1# x succ making the following diagram commute:

UxsuccA x co-> A x co

*l^xsucc

*

B x co-^ B x co


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70 PAULCORAZZA

For each A e V and each w#: B ? B G V?, we obtain a bijection

?a,uB:nomvo(F(A),uB)9*Homv(A,G(uB))

natural inA and It isconvenient to define the inverse ap ?~^u .Givenf:A?> G(ub) = B, we seek a unique (f>: Ia x succ ?> w^, as in the

following:iA XSUCC

A x co->- A x co

e

B x co->- B x co

Defining xq\ A A x co by a i?> (a,0), we can let be the uniquemap h guaranteed by Freyd's Theorem. It is routine to verify that thiscorrespondence isa bijection that is natural inA and H

As we observed in the proof, the critical point of the Lawvere functorj: V ?> V is 1, and j(l) = co. The seed behavior of j is given by thefollowing:

2.3. Lawvere Seed Proposition. Working in ZFC, let j = G o F be the

Lawvere functor. Then 0 e G(co co) is a universal element for G. Inparticular, for every endo B B and every y G G{ub)

= B there is a uniquea: succ ?> ub so that G (a)(0)

? y. Moreover,

V = {G(a)(0) | isan arrow inV^}.Proof. Let x: 1?> co: 0 ?->0. Identify the element >>G G(wjb) with the

map 1? 2?: 0 i->>>,and identify F(l) with succ via the canonical isomor

phism. Then the universality f the component x = r\\: 1?

G{F{\)) of

the unit rj of the adjunction F -\ G gives us as before a unique a: succ ?> ubso that the following triangle commutes:

succ 1?^GF(l)a \ G(a)

G{?B)The

equation G{a)ox ?

y immediately ivesthe desired conclusion. H

The Trnkova-Blass Theorem. In this section we review some of the work

of Blass [2], showing that there sameasurable cardinal if nd only if thereisan exact functor from V to V not (naturally) isomorphic to the identity(the reader isalso referred oTrnkova's work [32]for somewhat different

argument). In some cases we have re-formulated his results to emphasizethe framework of generalization that we pursue in this paper; we include


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j: V ?> V 71

additional discussion or proofs for these points of departure when neces

sary. An example of such a re-formulation is our omission of the naturalitycondition in the original statement of the Trnkova-Blass Theorem. Thisomission was discussed in Part

1,and the

proofof its harmlessness is

givenin the comments following Theorem 2.12.We begin with several definitions nd results that we statewithout proof

(proofs an be found in [2]).As was done for awvere's Theorem, we treat V(amodel of ZFC or even ZFC - Infinity) s the category of sets. Recallthat a functor is left xact if it preserves all finite limits; equivalently, f itpreserves all finite products and all equalizers. Dually, a functor is rightexact if it preserves all finite co-limits; equivalently, if it preserves all finite

co-products and all co-equalizers.A helpful observation (whichwas also made in Part 1) is that, ince every

set is (isomorphic to) a coproduct of l's, if a functor preserves 1 (as itmust if it preserves all finite limits) and all set-indexed coproducts, it isnaturally isomorphic to the identity unctor. s is the casewith elementaryembeddings V ?> M, ameasurable cardinal "tends to" arise as the first breakin this type f symmetry: f is the east cardinal forwhich F(k) ^ k (where

F is exact), k,will turn out inmany cases to be measurable, and will alwaysbe at least as big as the least measurable (whichmust exist). Conversely,given ameasurable cardinal one can define an exact functor F such that kis the critical point ofF.

Notice that, ifF: V ?> V is an exact functor that is not isomorphic tothe identity, hen if is any set for which F(X) ^ X, it follows that X isinfinite sinceF preserves all finite oproducts of 1). This observation canbe established in the theory FC ? Infinity, nd allows us to conclude thattheAxiom of Infinity sprovable from

ZFC ?Infinity

"there is an exact functor romV to Vnot isomorphic to the identity."

We will often state in hypotheses of theorems that "F is an exact functor"; to be precise, we think of F as a class defined possibly with parameters. A theorem that assumes the existence of such a functor shouldbe viewed as a schema of theorems, one for each class-defining formulaF.

Suppose F: V ?> V is left xact and F(0) = 0. For any setA and any

a e F(A), defineFA,a(X) = {F(f)(a)\f:A-+X}.

Formally, from formula that defines , we have a finitistic rocedure forobtaining a formula defining FAta, with extra parameters A, a.

SupposeD isa filter n a nonempty setA.lff and g are partial functionson A, we write / ~ g if nd only if the set of a forwhich / and g are both


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72 PAULCORAZZA

defined and equal at a belongs to D. Define a functor Z>-Prod by

Z)-Prod(jr) = XA/D=

{[f] Iisa

partialfunction A X with domain in

D}.For any A: X ?> Y9

D-Vrod(h): D-Fvod(X) -+ D-Prod(r): [g] [hog].D-Pvod is called a reduced power mod D. It is straightforward to show thatZ>-Prod is a left exact functor. We state the next several lemmas without

proof:2.4. Subfunctor Lemma. Suppose F: V ?> V is left exact andF($) = 0.

SupposeX is a set.

(1)For everyA9a9FAta(X) QF(X).(2)For every GF(X), x GFyfJC(JSr).(3) For every A9 a9 FA,a :V ?> V is left exact.

(4) Suppose I is set. Then F preserves ll I-indexed coproducts f nd onlyif a,q preserves all I-indexed coproducts, for all A9 a. In fact9 for anycollection {Bi \ G /},

^(U*)-II*wi iif nd only if or eachA9a

FA^jlB^^llFAABi).i iParts (1)and (2)of theLemma imply hat, for ach X, F{X) is the union

over all A9 a of F/4,a(Ar). Given a left exact F: V ?> V9 suppose A is a setand a GF{A). Define a filter over A by

D = {XQA\a ? ranFfo)},where ix: X A is the inclusion map. Z> is the filter derived from F9A9a.Notice that this sessentially hefamiliar efinition f ameasurable ultrafilterfrom an elementary embedding: if happens to preserve the subset relation

(moreprecisely, ll inclusionmaps), the xpression a G ranF(ix)" becomesthemore familiar "a GF(X)".

2.5. Equivalence Lemma. SupposeF \V -* V is left xactwith (0) = 0.Let A be a nonempty et nd a GF{A). Let D be the ilter derivedfrom 9A9a.Then

Fa,u is naturally isomorphic to Z>-Prod.

2.6. Reduced Power Filter Lemma. Suppose D is afilter.

(1)D-Prod preserves inite coproducts f nd only ifD isan ultrafilter.(2) Suppose X is an infinite cardinal D-Prod preserves X-indexed coproducts

if nd only if very ntersection f X elements fD is lso in .


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the axiom of infinity and transformations j : V -> V 73

2.7. Coproduct Coequalizer Lemma. Suppose F: V ?> F w fe/f exac/1with (0) = 0. F preserves coequalizers if nd only ifF preserves co-indexedcoproducts.

2.8. Nonprincipal Filter Lemma. Suppose n, Xare infinite cardinals andkis uncountable. Suppose D is an ultrafilter over X.

(1) IfD-Pvo&{k) ^ k, then isnon-principal.(2) IfX ? k andD isnonprincipal nd n-complete, thenD-Prod(ft) ^ k.Proof. For (1), assume D is principal with generator y < X. For each

a < k, let ca: X ?> k be the constant function with value a. Now observethat for ny total /: X?> k, if = f{y), then [/] = [ca].Also notice thatfor every partially defined g :X k, there is a total f:X?>k such that[g]= [/]. Therefore |Z)-Prod(?)| = ac, s required.

For (2), assume Z) is nonprincipal and suppose (fa: a < k) are suchthat for each a, [fa] isan element ofD-Prod(k) and if ^ /?, ^ [/p].

Moreover, WLOG, assume eachfa is total. Define g: k ?> /cby letting (/?)be the least element not belonging to {/ V that is not isomorphic to the identity. In

particular, k is the critical point of 2)-Prod.

Outline of Proof. As we remarked earlier, Z>-Prod is left exact. Since Dis an ultrafilter, Z>-Prod preserves finite coproducts. Since D is ^-completeand k is uncountable, Z>-Prod preserves countably indexed coproducts;hence, by the Coproduct Coequalizer Lemma, Z>-Prod preserves coequalizers. Therefore, Z>-Prod is right exact. Morever, by the Reduced Power FilterLemma and ^-completeness of D, Z)-Prod preserves all A-indexed coproducts, for X < k, and so Z)-Prod(A) = X for all such X. By the NonprincipalFilter Lemma, Z>-Prod(/s) ^ k, and so Z>-Prod is not isomorphic to theidentity; indeed, k is the critical point of Z)-Prod. H

2.10. Trnkova-Blass Theorem (Second Half). Suppose F :V ?> V isan exact class functor not isomorphic to the identity functor. Then V (="there exists a measurable cardinal". Indeed, ifk is the least cardinalfor which

F(k) ^ k, then, nV, k is t least s bigas the eastmeasurable (whichmust xist).Moreover, ifk is the eastelement f {a: a is the ritical oint of an exactfunctor V ?* V that is not isomorphic to the identity}, then k is the least measurable cardinal in V.

Outline of Proof. Let k be the smallest cardinal for which F(k) ^ k.By our earlier remarks, k must be infinite. By the Coproduct CoequalizerLemma, since F preserves coequalizers, k must be uncountable. By part (4)


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74 PAULCORAZZA

of the Subfunctor Lemma, since F does not preserve the coproduct k =

UK 1,for someA,a, FAm also fails to preserve \JK .Therefore, if is thefilter derived from FA, a,

Z)-Prod(tt) ^ FAm(k) ? ?.Since F preserves coequalizers, so must Z)-Prod, and by the Reduced

Power Filter Lemma, D isa>\-complete. By the Nonprincipal Filter Lemma,D must also be nonprincipal. It follows that there must exist a measurablecardinal in V (see for example [11, Corollary 2.12]). (Note that the entireargument takes place inside V since both F and FAm are definable over V.)

Let y denote the leastmeasurable cardinal in K. It follows that D is

y-complete (seefor

example [11,Theorem

2.11]). BytheReduced Power

Filter Lemma again, it follows that Z>-Prod preserves /^-indexed coproductsfor ll p < y. SinceD-Prod does not preserve ]JK 1, it follows that y < k.

Finally, let s suppose that k happens tobe the east element f {a: a is thecritical point of an exact functor V ?> V that is not isomorphic to the

identity}. We show that k = y. It suffices o show that k < y. Let Ube a nonprincipal y complete ltrafilter n y. By the proof of theFirst Halfof the Trnkova-Blass Theorem, U-Prod is an exact functor V ?> V with

(/-Prod(y) ^ y. By leastness of k < y, as required. HWe remark here that thedefinition f n as the least element of the set =

{a: a is the critical point of an exact functor V ?> Fthat isnot isomorphicto the identity} sformally xpressible inZFC. Consider the following set,defined inZFC:

T = {a : 3A, D (D is an a>\-complete nonprincipal ultrafilter

on A and a is least such that -Vrod(a) ^ a)}.We show k is the least element of T. To begin, we let denote the

least element of T with witnesses A,D, so that Z)-Prod(tto) ^ /^o- SinceD-Prod: V ?> V is exact, k < kq. Conversely, let F: V

? V be an exactfunctor witnessing membership of k in S. Since F(k) ^ then, as we ar

gued before, theremust be A, a such that Fa^(k) ^ k and a correspondingD-Prod such that D is coi-complete and D-Prod(k) ^ k. Therefore, kq < k.

We have shown k is the least element of T.We also mention here that the proof of Trnkova-Blass Theorem (Second

Half) makes essential use of the fact that isa class over V. In order for

the derived filter to be a set in V, F must be given by a formula. Later,we will discuss other functors that are not definable over V and itwill beclear that the analogue toD will not provide us with ameasurable cardinalin V. (A rough example along these lines that can be given at this point isan elementary embedding : L

? L with critical point as,where L is theconstructible universe and j is given by the existence of 0#. Though onecan define (in V) the subcollection D = {X G Pl(k) |k G j{X)} of L,


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THE AXIOM OF INFINITY AND TRANSFORMATIONS j :v ?> v 75

D is not a set inL precisely because it is defined in terms of the externalembedding .)

In the proof ofTrnkova-Blass Theorem (SecondHalf), onemight hope to

prove that the critical point of any exact functor V ?> V ismeasurable. Theobvious strategy to accomplish this is to observe, as in the proof, that, givenan exact F: V ?> V, not isomorphic to the identity, there is, over some A,an co -complete nonprincipal ultrafilter forwhich Z)-Prod(Ac) k (wherek is the critical point of F). By leastness of we have that F preservesA-indexed coproducts of 1 for all uncountable X < k. The natural hopeis that F preserves all A-indexed coproducts (inwhich case, D would be/^-complete), but this is not generally the case, as A. Blass pointed out to theauthor. We formulate the issue as an

open question:2.11. Open Question. Suppose F: V ?> V is exact and not isomorphic

to the identity. Without assuming other conditions on F, is it necessarilytrue that the critical point ofF isa large cardinal?

As we mentioned in Part 1, our version of the Trnkova-Blass Theoremfollows easily from the original theorems which referred to "natural iso

morphism" instead of merely "isomorphism" in the theorem statement. Weprove this now.

2.12. Original TrnkovA-Blass Theorem [2, 32]. There is an exactfunctor V ? V not naturally isomorphic to the identity f and only ifthere is a measurable cardinal.

Assume the truth of Original Trnkova-Blass Theorem; we show how theversion used in this paper follows directly from the original formulation.Suppose there is an exact functor F: V

? V not isomorphic to the identity.Then F isnot naturally somorphic to the identity, nd soby the "Original"version, there is a measurable cardinal.

Conversely,if there is a measurable

cardinal, then the proof given above of the "First Half" of the Trnkova-BlassTheorem is identical to the original proof, which establishes the existence ofan exact functor not naturally isomorphic to the identity. The proof in factshows that if D is a ^-complete nonprincipal ultrafilter over a measurablecardinal Z)-Prod: V ?> V is an exact functor with Z)-Prod(^) ^ whichdemonstrates the existence of an exact functor not isomorphic to the identity,as required.

As our proof of theTrnkova-Blass Theorem indicates, t is often the casethat the critical point of an exact functor : V ? V that s not isomorphicto the identity is a measurable cardinal k. We show, however, that sucha functor is unable to generate all sets in V with seed k; in other words,V 7^{j'(/)(?) |dom / = k}. The reason is summarized in the followingProposition: First, if such a j does have a weakly universal element atall, j must be naturally isomorphic to one of the reduced product functorsD-Prod. Also, whenever D is co\ complete, the class \JX -Prod(X) is


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76 PAULCORAZZA

the usual well-founded ultrapower VK/D whose transitive collapse M doesnot contain D. In particular, when D is normal over k, ifwe identify the

ultrapower with its transitive collapse in the usual way, k, itself is the seed,

and we haveD ^ 7 (/)(?) for ny / forwhich dom f?

n.2.13. Restricted Seed Proposition. Suppose j: V ?> V isan exact func

tor not isomorphic to the identity.

(1) If j isnaturally somorphic o reduced ower Z>-Prod, then admits aweakly universal element.

(2) If j admits a weakly universal lement, then isnaturally somorphic oa reduced power.

(3) If D is a normal measure on k, k g Z)-Prod(ft) is weakly universal

for D-Prod {via theusual identification fD-Prod(n) with its transitivecollapse), but or some set Y, Y ^ Z>-Prod(/)(?) for any f.

Proof of (1). We show that for any reduced power D-Prod, the elementa = [id] g D-Prod(A) (whereA = \JD) isweakly universal for D-Prod:Suppose [/] g D-Prod(Ar) = XA/D. Without loss of generality, ssume/: A ?> X is total; but now / isa candidate towitness weak universality:

We must verify that Z)-Prod(/)([id]) = [/]; but this follows immediatelyfrom the definition of Z>-Prod on functions.

Proof of (2). Suppose F = j: V -? V is exact, not isomorphic to theidentity, nd has a weakly universal element a g F(A). We show that

F = FA,a- By theEquivalence Lemma, this suffices oprove (2). Supposex g F(X). Byweak unive

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The Axiom of Infinity

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