Explicit Mathematics With The Monotone Fixed Point Principle. II: Models Michael Rathjen Department of Pure Mathematics University of Leeds England Abstract This paper continues investigations of the monotone fixed point principle in the context of Feferman’s explicit mathematics begun in [14]. Explicit mathematics is a versatile formal framework for representing Bishop-style constructive mathematics and generalized recursion theory. The object of investigation here is the theory of explicit mathematics augmented by the monotone fixed point principle, which asserts that any monotone operation on classifications (Feferman’s notion of set) possesses a least fixed point. To be more precise, the new axiom not merely postulates the existence of a least solution, but, by adjoining a new constant to the language, it is ensured that a fixed point is uniformly presentable as a function of the monotone operation. Let T 0 + UMID denote this extension of explicit mathematics. [14] gave lower bounds for the strength of two subtheories of T 0 + UMID in relating them to fragments of second order arithmetic based on Π 1 2 comprehension. [14] showed that T 0 „ +UMID and T 0 „ +IND N + UMID have at least the strength of (Π 1 2 - CA) „ and (Π 1 2 - CA), respectively. Here we are concerned with the exact reversals. Let UMID N be the monotone fixed-point principle for subclassifications of the natural numbers. Among other results, it is shown that T 0 „ + UMID N and T 0 „ +IND N + UMID N have the same strength as (Π 1 2 - CA) „ and (Π 1 2 - CA), respectively. The results are achieved by constructing set-theoretic models for the aforemen- tioned systems of explicit mathematics in certain extensions of Kripke-Platek set theory and subsequently relating these set theories to subsystems of second order arithmetic. 1 Introduction The present paper is a continuation of [14] and therefore familiarity with [14] will be presumed and the same notations will be used. The main results of [14] were stated as follows (cf. [14], Theorem 5.3): Theorem Let φ be a Π 1 3 sentence of second order arithmetic and φ * be its canonical translation into the language of explicit mathematics. (i) If (Π 1 2 - CA) „ ‘ φ, then T 0 „ +UMID ‘ φ * . (ii) If (Π 1 2 - CA) ‘ φ, then T 0 „ +IND N + UMID ‘ φ * . Inspection of the proofs in [14, 13] readily reveals that the interpretations do not require the principle UMID for the entire universe. A restricted version of UMID, 1
Transcript
Explicit Mathematics With The Monotone Fixed Point
Principle. II: Models
Michael RathjenDepartment of Pure Mathematics
University of LeedsEngland
Abstract
This paper continues investigations of the monotone fixed point principle in thecontext of Feferman’s explicit mathematics begun in [14]. Explicit mathematics is aversatile formal framework for representing Bishop-style constructive mathematicsand generalized recursion theory. The object of investigation here is the theory ofexplicit mathematics augmented by the monotone fixed point principle, which assertsthat any monotone operation on classifications (Feferman’s notion of set) possessesa least fixed point. To be more precise, the new axiom not merely postulates theexistence of a least solution, but, by adjoining a new constant to the language, it isensured that a fixed point is uniformly presentable as a function of the monotoneoperation. Let T0 +UMID denote this extension of explicit mathematics. [14] gavelower bounds for the strength of two subtheories of T0 +UMID in relating them tofragments of second order arithmetic based on Π1
2 comprehension. [14] showed thatT0 ¹ +UMID and T0 ¹ +INDN + UMID have at least the strength of (Π1
2−CA)¹and (Π1
2 −CA), respectively.Here we are concerned with the exact reversals. Let UMIDN be the monotone
fixed-point principle for subclassifications of the natural numbers. Among otherresults, it is shown that T0 ¹+UMIDN and T0 ¹ +INDN + UMIDN have the samestrength as (Π1
2 −CA)¹ and (Π12 −CA), respectively.
The results are achieved by constructing set-theoretic models for the aforemen-tioned systems of explicit mathematics in certain extensions of Kripke-Platek settheory and subsequently relating these set theories to subsystems of second orderarithmetic.
1 Introduction
The present paper is a continuation of [14] and therefore familiarity with [14] will bepresumed and the same notations will be used.
The main results of [14] were stated as follows (cf. [14], Theorem 5.3):
Theorem Let φ be a Π13 sentence of second order arithmetic and φ∗ be its canonical
translation into the language of explicit mathematics.
(i) If (Π12 −CA)¹` φ, then T0 ¹ +UMID ` φ∗.
(ii) If (Π12 −CA) ` φ, then T0 ¹ +INDN + UMID ` φ∗.
Inspection of the proofs in [14, 13] readily reveals that the interpretations do notrequire the principle UMID for the entire universe. A restricted version of UMID,
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termed UMIDN, suffices. UMIDN asserts that any monotone operator f which sendssubclassifications of N to subclassifications of N has a least fixed point lfp(f). Thus [14]Theorem 5.3 can be sharpened as follows:
Theorem 1.1 Let φ be a Π13 sentence of second order arithmetic.
(i) If (Π12 −CA)¹` φ, then T0 ¹ +UMIDN ` φ∗.
(ii) If (Π12 −CA) ` φ, then T0 ¹ +INDN + UMIDN ` φ∗.
The intent of this paper is to show that the reversals of the above theorem hold too.They are obtained by carefully constructing set-theoretic models for the aforementionedsystems of explicit mathematics in certain extensions of Kripke-Platek set theory andsubsequently relating these set theories to subsystems of second order arithmetic. To bemore precise, we construct a class model M of T0 ¹ +INDN + UMIDN in KPw+Σ1-Sep.The latter system is Kripke-Platek set theory (which is taken to include the axiom ofinfinity) augmented by Σ1 Separation with Foundation restricted to sets but retaininginduction on ω for all formulae. The model M arises as a union of (set) models MVn,hn .For meta n, the existence of MVn,hn can already be proved in KPr+Σ1-Sep which is thetheory KPw +Σ1-Sep shorn of the schema of induction on ω. With regard to the theoryT0 ¹ +UMIDN we then show that those theorems of T0 ¹ +UMIDN, wherein all theclassification quantifiers are existential, hold in any of the models MVn,hn . The latterreduces T0 ¹ +UMIDN to KPr + Σ1-Sep. Since KPr + Σ1-Sep and KPw + Σ1-Sepprove the same theorems of second order arithmetic as (Π1
2 − CA) ¹ and (Π12 − CA),
respectively, the direction “⇐” in Theorem 1.1 ensues.There are some indications that M might actually be a model of UMID. But a
proof seems to require more “fine structure theory” of M than is provided in the presentpaper. However, we pursue the question of how much of UMID is actually modelled byM at some length. It is proved that M satisfies the monotone fixed point principle ona subuniverse I. This stronger principle, dubbed UMIDI, asserts the monotone fixedpoint principle for subclassifications of a classification I which comprises N and containsall the standard constants and is also closed under application.
The paper is organized as follows: Section 2 introduces restrictions of UMID and theaxioms pertaining to the classification constant I, and further describes the subsystemsof set theory, KPr + Σ1-Sep and KPw + Σ1-Sep, needed for the interpretations.
Section 3 is devoted to the construction of the model M, yielding the reduction ofT0 ¹ +INDN + UMIDN to KPw + Σ1-Sep.
Section 4 is concerned with the reduction of T0 ¹+UMIDN to KPr +Σ1-Sep. Sincethe full model M cannot be formalized in T0 ¹ + UMIDN, the approach is to first showpartial cut elimination for T0 ¹ + UMIDN in a Tait-style calculus and subsequently in-terpret partially normalized derivations of ΣEM formulae in the approximations MVn,hn .The ΣEM formulae are those wherein all the classification quantifiers are existential.
The circle of reductions is completed in Section 5 by showing that KPr + Σ1-Sepand KPw +Σ1-Sep prove the same theorems of second order arithmetic as (Π1
2−CA)¹and (Π1
2 −CA), respectively.
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2 Preliminaries
2.1 Variants of the monotone fixed point principle
Definition 2.1 As in [14] we use the abbreviations X◦= Y := ∀v(v
◦∈ X ↔ v◦∈ Y )
and X◦⊆ Y := ∀v(v
◦∈ X → v◦∈ Y ). To state the monotone fixed point principle for
subclassifications of a given classification A we introduce the following shorthands:
Clop(f ,A) if ∀X ◦⊆ A ∃Y ◦⊆ A fX ' Y
Ext(f ,A) if ∀X ◦⊆ A ∀Y ◦⊆ A [X ◦= Y → fX◦= fY ]
Mon(f,A) if ∀X ◦⊆ A ∀Y ◦⊆ A [X◦⊆ Y → fX
◦⊆ fY ].
Lfp(Y, f,A) if fY◦⊆ Y ∧ Y
◦⊆ A ∧ ∀X ◦⊆ A[fX
◦⊆ X → Y◦⊆ X
]
When f satisfies Clop(f ,A), we call f a classification operation on A. When f satisfiesClop(f ,A) and Ext(f ,A), we call f extensional or an extensional operation on A. Whenf satisfies Clop(f ,A) and Mon(f,A), we say that f is a monotone operation on A. Sincemonotonicity entails extensionality, a monotone operation is always extensional.
Now we state UMIDA.
UMIDA (Uniform Monotone Inductive Definition on A)
∀f [Clop(f ,A) ∧Mon(f, A) → Lfp(lfp(f), f, A)].
UMIDA states that if f is monotone on subclassifications of A, lfp(f) is a least fixedpoint of f .
We also introduce an extension of T0, dubbed T0(I), which has an additional classi-fication constant I and axioms pertaining to I, asserting that I is a classification whichis closed under application and contains all the basic constants, i.e. e
◦∈ I where e is anyof the constants 0, sN,pN,k, s,d,p,p0,p1, i, j, cφ,N, lfp, I, and
∀x ◦∈ I ∀y ◦∈ I ∀z (xy ' z → z◦∈ I ).
Since T0(I)¹ ` N ◦⊆ I, we have T0 ¹ +UMIDN ⊆ T0(I)¹ +UMIDI.
2.2 Subsystems of set theory
The axiom systems for set theory considered in this paper are formulated in the usuallanguage of set theory (called L∈ hereafter) containing ∈ as the only non-logical symbolbesides =. Formulae are built from prime formulae a ∈ b and a = b by use of propositionalconnectives and quantifiers ∀x,∃x. Bounded quantifiers ∀x ∈ a, ∃x ∈ a are defined asusual. ∆0-formulae are the formulae wherein all quantifiers are bounded; Σ1-formulaeare those of the form ∃xϕ(x) where ϕ(a) is a ∆0-formula. For n > 0, Πn-formulae (Σn-formulae) are the formulae with a prefix of n alternating unbounded quantifiers startingwith a universal (existential) one followed by a ∆0-formula. The class of Σ-formulae isthe smallest class of formulae containing the ∆0-formulae which is closed under ∧, ∨,bounded quantification and unbounded existential quantification.
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The exact details of the formulation do not really matter for the purpose of thispaper, any standard formulation will work. Also, we use the standard ∆0-definitions ofpredicates like x = ∅, Tran(x), On(x) and the like. In what follows we shall assumefamiliarity with the basics of admissible set theory as presented in [3], Chap. I,II.
Definition 2.2 We use Kripke-Platek set theory KP (cf. [3]) as our basic theory. Itconsists of the axioms of Extensionality, Pairing, Union, Infinity1 and of the axiomschemata of Separation and Collection for ∆0-formulae as well as the Foundation schemafor arbitrary formulae.
KPr arises from KP by replacing the axiom schema of Foundation by the Foundationaxiom
∀x(∃y(y ∈ x) → ∃y(y ∈ x ∧ ∀z ∈ x(z /∈ y))).
KPw is obtained from KPr by adding the schema
INDω ∀x ∈ ω(∀y ∈ xφ(y) → φ(x)) → ∀x ∈ ωφ(x)
of induction on ω to KPr (for all formulae φ).
Definition 2.3 Σn-Separation (abbreviated Σn-Sep) is the schema of axioms
∃z ∀u (u∈z ↔ [u∈a ∧ φ(u)]
)
for all set-theoretic Σn-formulae φ with z not free in φ.
2.3 Some results derivable in KPr + Σ1-Sep
For later use we collect some derivable consequences of KPr + Σ1-Sep.
Definition 2.4 The schema of Σ Foundation consists of all formulae
∀x [(∀y ∈ x)φ(y) → φ(x)] → ∀xφ(x),
where φ is a Σ formula.
Proposition 2.5 All instances of Σ Foundation are provable in KPr + Σ1-Sep.
Proof : First, we show that KPr + Σ1-Sep proves that every set a has a transitiveclosure TC(a). For n ∈ ω let ψ(n, f) be the formula expressing that f is a functionwith domain n + 1 such that f(0) = a ∧ (∀k < n) f(k + 1) = f(k) ∪⋃
f(k). The classA := {n ∈ ω : ∃fψ(n, f)} is a set by Σ1 Separation. Using the axiom of Foundation,one easily proves that A = ω. Moreover, (∀n ∈ ω)∃!y∃f [ψ(n, f) ∧ ran(f) = y]. Thus,by Σ Replacement (cf. [3], I.4.6) there exists a function F with domain ω such that(∀n ∈ ω)∃f [ψ(n, f) ∧ ran(f) = F (n)]. Obviously, we have TC(a) = ran(F ).
Now, for a contradiction, suppose we have a failure of Σ Foundation. Then
∀x [(∀y ∈ x)φ(y) → φ(x)]
but ¬φ(a) for some a. Set B := {y ∈ TC({a}) : ¬φ(y)}. Since every Σ formula isequivalent to a Σ1 formula (using ∆0 Collection), C := {y ∈ TC({a}) : φ(y)} is a set byΣ1 Separation. As B = TC({a}) \ C, B is a set by ∆0 Separation. Observe that B 6= ∅since a ∈ B. Using the Foundation axiom there exists c ∈ B such that (∀z ∈ c)(z /∈ B).Since TC({a}) is transitive the latter implies (∀z ∈ c)φ(z), yielding φ(c). But thatcollides with c ∈ B. 2
1For the results of this paper it is crucial that Infinity is assumed to be among the axioms of KP.
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Lemma 2.6 In KPr +Σ1-Sep we can define functions by Σ Recursion. To put it moreformally, if Φ(~x, y, z) is a Σ formula, then
where Fun(f) asserts that f is a function and dom(f) denotes its domain.
Proof : As far as Foundation is concerned, the proof of Σ Recursion in KP only requiresΣ Foundation as can immediately be gleaned from the proof of [3], I.6.4 or [5], I.11.8. 2
As a consequence, all the familiar functions defined via Σ Recursion are at our disposalin KPr + Σ1-Sep. Prominent examples are the rank function, the Mostowski collapsingfunction as well as the constructible hierarchy.
Definition 2.7 We will use Godel’s constructible hierarchy L = (Lα)α∈On in one of itsusual formulations. For definiteness let
L0 = ∅, Lα+1 = Def(Lα), Lλ =⋃
α<λ Lα for λ ∈ Lim.
Here Def(x) is the set of all definable subsets of x.For subsets U ⊆ ω we will also consider the relativized constructible hierarchy L(U) =⋃
α∈On Lα(U) which is defined as follows:
L0(U) = ∅, Lα+1(U) = DefU (Lα(U)), Lλ(U) =⋃
α<λ Lα(U) for λ ∈ Lim.
Here DefU (x) is the set of all subsets definable over the structure (x,∈¹ x2, U ∩x) in thelanguage L∈(R) which contains an additional relation symbol R.
Definition 2.8 Let r ⊆ V × V . For a set x let rx := {y : 〈y, x〉 ∈ r}. Set
wfp(a, r) :=⋂{u ⊆ a : ∀x∈a [{rx ⊆ u → x∈u],
i.e. the well-founded part of r on a. r is said to be well-founded on a if if wfp(a, r) = aholds or, in equivalent terms, if
∀b [b ⊆ a ∧ b 6= ∅ → ∃x∈b∀y∈b (〈y, x〉/∈r)
].
r is well-founded if r is well-founded on a for (any) some a satisfying r ⊆ a× a .
Lemma 2.9 The function (a, r) 7→ wfp(a, r) is Σ1 in KPr + Σ1-Sep.The predicate of being well-founded is provably ∆1 in KPr + Σ1-Sep.
Proof : Let us work in KP + Σ1-Sep. We write x≺y for 〈x, y〉 ∈ r. Define an operationF on the ordinals by Σ Recursion (Lemma 2.6):
F (α) = {x ∈ a : ∀y∈a (y≺x → ∃β < α y∈F (β))}= the set of all x∈a such that {y∈a : y≺x} ⊆
⋃
β<α
F (β).
Note that β ≤ α implies F (β) ⊆ F (α). There is a set
b0 = {x∈a : ∃α (x∈F (α))},
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this being the place where we need Σ1 Separation. Note that if x∈a and rx ⊆ b0, then,by Σ Reflection, there exists α such that rx ⊆ F (α) and hence x∈F (α+1), showing thatx ∈ b0. On the other hand, if a set u ⊆ a satisfies ∀x∈a [rx ⊆ u → x∈u], then one easilyverifies by induction on α that F (α) ⊆ u; whence b0 ⊆ u. The upshot of the above isthat b0 = wfp(a, r).
To show that b0 has a Σ1 definition, observe first that by Σ Reflection there existsan ordinal ρ such that b0 = {x∈a : ∃α < ρ (x∈F (α))}. Thus the desired Σ1 definition ofwfp(a, r) is given by
b = wfp(a, r) ↔ ∃f ∃ρ [dom(f) = ρ + 1 ∧
⋃ran(f) = b ∧ ∀α ≤ ρ
f(α) = the set of all x∈a such that {y∈a : y≺x} ⊆⋃
β<α
f(β)
∧ f(ρ) =⋃
β<ρ
f(β).
Further, since the predicate b = wfp(a, r) is Π1 by definition, the above also yields thesecond assertion of the present lemma. 2
Corollary 2.10 (KP + Σ1-Sep) If r is well-founded on a and
∀x∈a [∀y∈a (〈y, x〉 ∈ r → φ(y)) → φ(x)],
then ∀x∈aφ(x).
Proof : By the previous lemma, we can pick a function f whose domain is an ordinalρ + 1 such that
⋃β<ρ f(β) = wfp(a, r) and
∀α ≤ ρ f(α) = the set of all x∈a such that {y∈a : y≺x}.
¿From ∀x∈a [∀y∈a (〈y, x〉 ∈ r → φ(y)) → φ(x)] one then deduces
∀α ≤ ρ [∀β < α ∀y∈f(β)φ(y) → ∀x∈f(α)φ(x)].
Whence, using Foundation, it follows ∀α ≤ ρ∀x∈f(α)φ(x), and thus ∀x∈wfp(a, r)φ(x).2
2.4 Projection functions
In this section we assume some results from recursion theory on admissible sets, inparticular results about projectibility and characterization of stable ordinals. Detailscan be found in [3].
To begin with we recall some definitions from ordinal recursion theory.
Definition 2.11 An ordinal κ is said to be stable if Lκ ≺1 L, i.e. Lκ is a Σ1-elementarysubstructure of L.
Let ρ > κ. κ is ρ-stable if Lκ ≺1 Lρ.Another rendering of stability comes in terms of ordinal recursion theory (cf. [11],
VIII.5.1):
κ is stable iff κ is closed under all ∞-partial recursive ordinal functions.
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Likewise,
κ is ρ-stable iff κ is closed under all (∞, ρ)-partial recursive functions.
Definition 2.12 For n ∈ ω define
σ0 := {α : α is Σ1 definable in L without parameters}σn+1 := {α : α is Σ1 definable in L in the parameters σ0, . . . , σn}.
The existence of the ordinals σn follows via Σ Separation. (n 7→ σn)n∈ω is a definableclass function in KPw+Σ1-Sep whereas in KPr+Σ1-Sep we can only prove the existenceof σn for external n.
Lemma 2.13 Lσn ≺1 Lσn+1, i.e. Lσn is a Σ1 elementary substructure of Lσn+1.
Proof : [3], V.7.9. 2
Lemma 2.14 For each n there exists a function
fn : Lσn
1−1→ ω
which is Σ1 definable on Lσn with the aid of the parameters σ0, . . . , σn−1.
Proof : Combine [3], V.7.10 and [3], V.6.2. 2
Definition 2.15 Let ∂(k,m) := 52k·3m. The functions gn : Lσn
1−1→ ω are defined byrecursion on n as follows:
g0(x) = ∂(0, f0(x))
gn+1(x) ={
gn(x) if x ∈ Lσn
∂(n + 1, fn+1(x)) if x /∈ Lσn
g∞ :=⋃n∈ω
gn.
Lemma 2.16 (i) gn ⊆ gn+1.
(ii) g∞ :⋃
n∈ω Lσn
1−1→ ω.
(iii) ran(g∞) ⊆ {5k+1 : k ∈ ω}.
Proof : Obvious. 2
Remark 2.17 The existence of the functions gn can be proved in KPw + Σ1-Sep.(n 7→ gn)n<ω is a definable class function of KPw + Σ1-Sep. Consequently, g∞ is adefinable class function of KPw + Σ1-Sep, too.
In the case of KPr +Σ1-Sep the functions gn can be proved to exist for meta n only.
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2.5 Relativizations
We will also have use for relativizations of the ordinals and functions of the previoussubsection with regard to a set U ⊆ ω.
Definition 2.18 For n ∈ ω define
σU0 := {α : α is Σ1 definable in L(U) without parameters}
σUn+1 := {α : α is Σ1 definable in L(U) in the parameters σU
0 , . . . , σUn }.
fUn ,gU
n , and gU∞ are defined in a similar vein.
The relativizations of the results of the previous subsection can be proved in the sameway and in the same theories as their unrelativized cousins.
3 Models
3.1 Applicative structures
Models for the applicative part of T0 (dubbed applicative structures) can be developedin very weak systems of set theory, since only recursively enumerable sets are required.The basic method for building models of T0 upon applicative structures goes back to [6,104–107]. Here we shall extend a particular model construction over a “free” applicativestructure introduced in [7, 3.3].
Since the models we use are already well described in the literature, cf. [7, 8, 16], wedo not present the full details.
We start off with the pairing structure Spair = (S, π, π0, π1, 0), where S = ω andπ : S2 → S \ {0} is an injective (recursive) pairing function with (recursive) inversesπ0, π1 such that π0(0) = π1(0) = 0. For technical reasons we moreover fix a special suchfunction π, namely π(x, y) = 2x · 3y. As its inverses, we fix π0, π1 where π0(z) = x andπ1(z) = y if z = 2x · 3y and π0(z) = π1(z) = z if z cannot be written in this form.
We call the base set S (and not ω) since we will have other “natural numbers” in thismodel and we want to avoid confusion between those two sets. Moreover, the intuitionabout S is that S consists of general objects and not only of the natural numbers.
For each n ∈ ω, the representation n◦ ∈ S of n in the structure Spair is defined in-ductively by 0◦ = 0, (n+1)◦ = π(0, n◦). The classification constant N will be interpretedas the set of all n◦, where n ∈ ω. More generally, for X ⊆ ω let X◦ = {n◦ : n ∈ X}. Inthe following, we use the codes
0 = 0,k = 1◦, s = 2◦,p = 3◦,p0 = 4◦,p1 = 5◦,d = 6◦, sN = 7◦,pN = 8◦,i = 9◦, j = 10◦, lfp = 11◦,N = 12◦, I = 13◦,U = 14◦, and cm = (15 + m)◦.
The relation AppS ⊆ S3 is inductively defined by the following clauses, wherewe use the abbreviations xy ' z :≡ AppS(x, y, z), (x, y) for π(x, y) and inductively(x1, x2, . . . , x3+n) := (x1, (x2, . . . , x3+n)).
• kx ' (k, x), (k, x)y ' x
• sx ' (s, x), (s, x)y ' (s, x, y).
• If xz ' u, yz ' v and uv ' w, then (s, x, y)z ' w.
Sapp := 〈S,AppS,k, s,p,p0,p1,d, sN,pN,0,N, I, (cm)m∈ω, i, j, lfp〉
such that Sapp models the applicative part of T0. Sapp can be shown to be an elementof LωCK
1, actually, it can be shown to be coded by recursively enumerable sets.
Definition 3.1 a) Let B ⊆ S be defined as B := S \ (π[S2] ∪ {0}). Denoting theclosure of B under π by Gen(B), we see that π : S2 → S \B, Gen(B ∪ {0}) = Sand π0(x) = π1(x) = x for all x ∈ B. We say that B is an atomic base for S.
b) For x ∈ S let suppB(x) ⊆ B be defined by recursion on the definition of Gen(B ∪{0}) by suppB(x) = {x} for x ∈ B, suppB(0) = ∅ and suppB(π(x, y)) =suppB(x) ∪ suppB(y).
c) LetAut(B) := {σ : σ : B → B and σ is bijective}.
For F ⊆ B let
Aut(B/F ) := {σ ∈ Aut(B) : (∀x ∈ F )(σ(x) = x)}
Each σ ∈ Aut(B) induces a mapping σ : S → S via σ ¹ B = σ, σ(0) = 0, andσ(π(x, y)) = π(σ(x), σ(y)). Since σ is uniquely determined by σ, we shall identifyσ with σ.
Lemma 3.2 a) If xy ' z, then suppB(z) ⊆ suppB(x) ∪ suppB(y).b) If σ ∈ Aut(B), then xy ' z ⇔ σ(x)σ(y) ' σ(z).c) If σ ∈ Aut(B/F ) and suppB(x) ⊆ F , then σ(x) = x.
Proof : a), b), c) can be proved by induction over the definition of AppS. For detailscf. [7, 3.3] or [16]. 2
3.2 Iterating operators along well-founded relations
The inductive model construction to be introduced in the next subsection involves aclause which asks whether an operator can be iterated along certain well-orderings toproduce a fixed point. Here we provide the pertinent terminology.
For each elementary φ(a, b1, . . . , bm, A1, . . . , An) we shall write
Given a family (A, g) of classifications over A, i.e. gx is a classification for all x◦∈ A, we
set ⋃{gx : x
◦∈ A} := z.∃x (x, z)◦∈ B,
where B ' j(A, g).Let PdN(R, a) := y.(y, a)
◦∈ R ∧ y◦∈ N. Let F be an application term such for every
operator Φ and every operation g which satisfies ∀y ◦∈ PdN(R, a)∃Y g(R, Φ, y) ' Y ,
F(R, Φ, a, g) '⋃{g(R, Φ, y) : y
◦∈ PdN(R, a)}∪ Φ(
⋃{g(R, Φ, y) : y
◦∈ PdN(R, a)}).
Definition 3.3 Using the recursion theorem (cf. [14], 2.2 or [6], 3.3), we find an appli-cation term itN such that
∀xyz itN(x, y, z) ' F(x, y, z, itN).
As a result, if R is well-founded and Φ is an operator, then itN(R, Φ, a) ↓ and
itN(R, Φ, a) '⋃{itN(R,Φ, a) : y
◦∈ PdN(R, a)} (1)
∪ Φ(⋃{itN(R, Φ, a) : y
◦∈ PdN(R, a)}).
Finally, letItN(R, Φ) :=
⋃{itN(R, Φ, x) : x
◦∈ N}.Note that itN and ItN are closed application terms of T0, and that (1) can be proved inT0.
3.3 Models for T0 and more
Definition 3.4 For λ a limit ordinal, let
well(α) := {R ∈ Lα : R is a well-ordering of a subset of ω};WELL := {well(λ) : λ limit}.
In what follows variables X ,Y will always range over WELL.
In this subsection we work in KPr + Σ1-Sep . Starting from a set X ∈ WELL anda 1 − 1 mapping ` : X → B, a model MX ,` of T0 is built inductively above this givenfamily of well-orderings. In the model MX ,`, each set A ∈ X induces a classificationnamed by `(A). This method of constructing models of T0 originates with [6] and [8],however, it will be crucial for our purposes to extend the inductive definition of thosemodels by a new clause (iv) to produce least fixed points for operators.
Definition 3.5 By induction on α we define structures
MX ,`α = 〈S,CLX ,`
α ,◦∈X ,`
α ,AppS,k, s,p,p0,p1,d, sN,pN,0,N, I, (cm)m∈ω, i, j, lfp〉
extending Sapp. So we only have to define CLX ,`α ⊆ S and
◦∈X ,`
α ⊆ S×CLX ,`α .
10
a) CLX ,`0 = {N, I} ∪ ran(`).
z◦∈X ,`
0 N :⇔ z ∈ S◦.z◦∈X ,`
0 I iff z is in the closure of S◦ under π (i.e. z ∈ Gen(S◦)).For R ∈ X and z ∈ S set
z◦∈X ,`
0 `(R) :⇔ ∃n,m ∈ ω [nRm ∧ z = (n◦,m◦)].
b) If α = β + 1 is a successor, then let CLX ,`β ⊆ CLX ,`
α and◦∈X ,`
β ⊆ ◦∈X ,`
α . In addition,new classifications at level α are generated by the following rules:
(i) If φ is an elementary formula with Godelnumber m, then let cm(~x,~a) ∈ CLX ,`α
for all ~x ∈ S and ~a ∈ CLX ,`β . Further define
z◦∈X ,`
α cm(~x,~a) :⇔ MX ,`β |= φ[z, ~x,~a].
(ii) If a ∈ CLX ,`β , f ∈ S and M
X ,`β |= ∀x ◦∈ a∃Y (fx ' Y ), then let j(f, a) ∈ CLX ,`
(iii) For a, b ∈ CLM0,β let i(a, b) ∈ CLM0,α and for z ∈ S let
z◦∈X ,`
α i(a, b) :⇔ ∀X ⊆ S(Prog(a, b, X) → z ∈ X)
where Prog(a, b, X) :≡
∀x ◦∈X ,`
β a(∀y ◦∈ S [(y, x)
◦∈X ,`
β b → y ∈ X] → x ∈ X).
(iv) Let f ∈ S. Suppose there is a well-ordering R of ω in X such that the followinghold:
– MX ,`β |= ∀x∃Y itN(`(R), f, x) ' Y .
– MX ,`β |= ∃Z ∃X [ ItN(`(R), f) ' Z ∧ f(Z) ' X ∧ X
◦⊆ Z].– For all S ∈ X if S <L R, then
(a) MX ,`β |= ∀x∃Y itN(`(S), f, x) ' Y
(b) MX ,`β |= ∃Z ∃X [ ItN(`(S), f) ' Z ∧ f(Z) ' X ∧ X
◦6⊆Z].
Then lfp(f) ∈ CLX ,`α and
z◦∈X ,`
α lfp(f) :⇔ MX ,`β |= z
◦∈ ItN(`(R), f).
c) If α is a limit ordinal, let CLX ,`α :=
⋃β<α CLX ,`
β and◦∈X ,`
α :=⋃
β<α
◦∈X ,`
β .
Finally, set CLX ,` :=⋃
α CLX ,`α and
◦∈X ,`:=
⋃α
◦∈X ,`
α . Let
MX ,` := 〈S,CLX ,`,◦∈X ,`
,AppS,k, s,p,p0,p1,d, sN,pN,0,N, I, (cm)m∈ω, i, j, lfp〉
Remark 3.6 (i) Regarding part (iv) of Definition 3.5,b), note that the well-orderingR is uniquely determined by being the <L-least satisfying the pertaining conditions.
11
(ii) Note that e◦∈X ,`
I holds if e is any of the constants k, s,p,p0,p1,d, sN,pN,0,N, I,(cm)m∈ω,i, j,lfp, or e ∈ S◦. Moreover, if x, y
◦∈X ,`I and xy ' z, then z
◦∈X ,`I.
Lemma 3.7 The following are provable in KPr + Σ1-Sep:
(i) For all α, well(α) is a set. The map α 7→ well(α) is Σ1.
(ii) MX ,` is a set.
(iii) If X = well(ρ), then the the map(λ 7→ Mwell(λ),`¹well(λ)
)λ<ρ
is Σ1 (in the parameter `).
Proof : (i): The class well(α) is ∆1 definable in KPr + Σ1-Sep by 2.9; thus, using∆1 Separation, KPr + Σ1-Sep proves that well(α) is a set. As a result, the functionα 7→ well(α) is Σ1.
(ii): First, notice that the function α 7→ 〈 ◦∈X ,`
α ,CLX ,`α 〉 is Σ1. This is immediate for
the clauses (i) and (ii) of Definition 3.5. For the clauses (iii) and (iv) which involve thewell-founded part of a relation on a set and the notion of being well-founded, respectively,we draw on Lemma 2.9.
The applicative part of MX ,` is clearly a set. CLX ,` is a set by Σ1-Sep since CLX ,` ={a ∈ S : ∃α (a ∈ CLX ,`
α ) }. Likewise,◦∈X ,`
is a set by Σ1-Sep as◦∈X ,`
= {〈x, a〉 ∈ S× S : ∃α (〈x, a〉 ∈ ◦∈X ,`
α ) }.(iii): For λ < ρ, put Xλ := well(λ) and `λ := ` ¹well(λ). CLXλ,`λ is a set by (i) and
(ii), and we can use Σ Replacement to obtain a function f : CLXλ,`λ → ON defined by
f(a) = the least α. a ∈ CLXλ,`λα .
As a result, there exists β such that CLXλ,`λ = CLXλ,`λ<β = CLXλ,`λ
β . Whence,
λ 7→ the least β.CLXλ,`λ<β = CLXλ,`λ
β
is Σ1. Therefore λ 7→ MXλ,`λ is Σ1 too. 2
Lemma 3.8 Let X ,Y ∈ WELL, X ⊆ Y, ` : X 1−1→ B, ℘ : Y 1−1→ B, and ℘ ¹ X = `.Then:
a) For all α, CLX ,`α ⊆ CLY,℘
α .b) If a ∈ CLX ,`
α , then for all x ∈ S, x◦∈X ,`
α a ⇔ x◦∈Y,℘
α a.
c) If α ≤ β and a ∈ CLX ,`α , then for all x ∈ S, x
◦∈X ,`
α a ⇔ x◦∈X ,`
β a.
d) If α ≤ β and a ∈ CLX ,`α , then for all x ∈ S, x
◦∈X ,`
α a ⇔ x◦∈Y,℘
β a.
Proof : a) and b) are proved simultaneously by induction on α. c) is proved by inductionon β. d) follows from a),b),c). As the proofs are routine, we restrict ourselves to themost interesting case in a),b). Let α = β + 1. Suppose a ∈ CLX ,`
α via clause (iv). Thena = lfp(f) for some f ∈ S and there exists a well-ordering R of ω in X such that thefollowing hold:
12
• MX ,`β |= ∀x∃Y itN(`(R), f, x) ' Y .
• MX ,`β |= ∃Z ∃X [ ItN(`(R), f) ' Z ∧ f(Z) ' X ∧ X
◦⊆ Z].
• For all S ∈ X if S <L R, then
(a) MX ,`β |= ∀x∃Y itN(`(S), f, x) ' Y
(b) MX ,`β |= ∃Z ∃X [ ItN(`(S), f) ' Z ∧ f(Z) ' X ∧ X
◦6⊆Z].
Let X = well(λ) and Y = well(λ′). Since X ⊆ Y, we have λ ≤ λ′. Hence, if S′ ∈ Yand S′ <L R, then S′ ∈ X since λ is a limit. Thus, using the induction hypothesis andnoting that `(R) = ℘(R), we obtain
a) MY,℘β |= ∀x∃Y itN(℘(R), f, x) ' Y .
b) MY,℘β |= ∃Z ∃X [ ItN(℘(R), f) ' Z ∧ f(Z) ' X ∧ X
◦⊆ Z].c) For all S ∈ X if S <L R, then
(i) MY,℘β |= ∀x∃Y itN(℘(S), f, x) ' Y
(ii) MY,℘β |= ∃Z ∃X [ ItN(℘(S), f) ' Z ∧ f(Z) ' X ∧ X
◦6⊆Z].
As a result of the above, a = lfp(f) ∈ CLY,℘α and thus, by induction hypothesis,
z◦∈X ,`
α a ⇔ MX ,`β |= z
◦∈ ItN(`(R), f) ⇔ MY,℘β |= z
◦∈ ItN(℘(R), f) ⇔ z◦∈Y,℘
α a.
2
Remark 3.9 In the following we may omit the indices α in the predicate x◦∈X ,`
α a as thepreceding lemma shows that the relation is independent of these parameters as long asa ∈ CLX ,`
α .
Corollary 3.10 Let X ,Y ∈ WELL, X ⊆ Y, ` : X 1−1→ B, ℘ : Y 1−1→ B, and ℘ ¹X = `.Then:
a) CLX ,` ⊆ CLY,℘.b) If a ∈ CLX ,`, then for all x ∈ S, x
◦∈X ,`a ⇔ x
◦∈Y,℘
a.
Convention. From now on, when considering a structure MX ,`, upper case lettersX,Y, Z, . . . will be understood to vary over CLX ,`; this will save space and is in keepingwith the syntax of T0.
Definition 3.11 The collection of ΣEM-formulae is the smallest set of T0(I)-formulaewhich contains the quantifier-free formulae and is closed under ∨,∧, ∀x,∃x, and existen-tial quantification over classifications.
Corollary 3.12 (ΣEM-persistence) Let X ,Y ∈ WELL, X ⊆ Y, ` : X 1−1→ B, ℘ :Y 1−1→ B, and ℘ ¹X = `. If ψ is a ΣEM-sentence (with parameters) whose classificationparameters are in CLX ,`, then
MX ,` |= ψ ⇒ MY,℘ |= ψ.
13
Proof : This is proved by induction on the generation of ψ. The atomic case followsfrom 3.10 and the fact that MX ,` and MY,℘ share the same applicative part. If ψ is ofthe form ψ0 ∧ ψ1 or ψ0 ∧ ψ1, then the assertion is immediate by induction hypothesis.If ψ is of the form ∃xψ0(x) or ∀xψ0(x), then the assertion follows from the inductionhypothesis and the fact that these quantifiers range over the same realm of objects in bothstructures. Finally, let ψ be of the form ∃Uψ0(U). Then there exists Z ∈ CLX ,` suchthat MX ,` |= ψ0[Z]. Then Z ∈ CLY,℘ and MY,℘ |= ψ follows by induction hypothesis.
2
Definition 3.13 Given Y ∈ WELL, ı : Y 1−1→ B, ~ : Y 1−1→ B, and σ ∈ Aut(B), we saythat σ is an isomorphism between MY,ı and MY,~, written
σ : MY,ı ∼= MY,~,
if ~ = σ ◦ ı, i.e., for all u ∈ Y, ~(u) = σ(ı(u)).
Proposition 3.14 If σ : MY,ı ∼= MY,~, then
MY,ı |= φ[~x, ~X] ⇔ MY,~ |= φ[σ(~x), σ( ~X)]
holds for all formulae φ(~u, ~U) of T0(I), ~x ∈ S, and ~X ∈ CLY,ı. The latter comprisesthat
CLX ,~ = {σ(Y ) : Y ∈ CLX ,ı }.
Proof : One first shows by induction on α:
a) X ∈ CLY,ıα ⇔ σ(X) ∈ CLY,~
α ;b) If X ∈ CLY,ı
α , then for all x,
x◦∈Y,ı
α X ⇔ σ(x)◦∈Y,~
α σ(X).
If α > 0, then the assertions follow immediately from the induction hypotheses. Theonly interesting case is α = 0. Let X ∈ CLY,ı
0 . If X = N, then σ(X) = X ∈ CLY,~0 and
x◦∈Y,ı
N⇔ σ(x)◦∈Y,~
σ(N) follows from the fact that for all x ∈ S◦, x = σ(x).Now suppose X = ı(R), where R ∈ Y. As σ(X) = σ(ı(R)) = ~(R), it follows
σ(X) ∈ CLY,~α . Further, since for all n,m ∈ ω, σ((n◦,m◦)) = (n◦,m◦), we get
Conversely, if σ(X) ∈ CLY,~0 and σ(X) 6= N, then σ(X) = ~(R) for some R ∈ Y. As
~ = σ ◦ ı and all maps are injective, this implies X = ı(R); thus X ∈ CLY,ı.As a result of the above, we have
a) X ∈ CLY,ı ⇔ σ(X) ∈ CLY,~;b) If X ∈ CLY,ı, then for all x, x
◦∈Y,ı
X ⇔ σ(x)◦∈Y,~
σ(X).
14
The desired assertion is now (straightforwardly) proved by formula induction on φ. Notethat if φ is an atomic formula other than t
◦∈ X, then this follows from Lemma 3.2,taking into account that MY,ı and MY,~ both have the same applicative structure. 2
Notation Let Y ∈ WELL and ı : Y 1−1→ B. If Z ⊆ Y we use the notation ı[Z] for theset {ı(a) : a ∈ Z}.
If ~x is a tuple x1, . . . , xn, σ(~x) stands for σ(x1), . . . , σ(xn).
Lemma 3.15 Let Z ⊆ Y ∈ WELL, ı : Y 1−1→ B and ~ : Y 1−1→ B. Further, supposethat ı ¹ Z = ~ ¹ Z and ı[Y \ Z] ∩ ~[Y \ Z] = ∅. Then, letting F := ı[Z], there existsσ ∈ Aut(B/F ), such that σ = σ−1 and
σ : MY,ı ∼= MY,~.
Proof : Define
σ(x) :=
x if x /∈ ı[Y \ Z] ∪ ~[Y \ Z]~(ı−1(x)) if x ∈ ı[Y \ Z]ı(~−1(x)) if x ∈ ~[Y \ Z]
Obviously, we have ∀x ∈ F σ(x) = x and σ = σ−1. Further, we have σ ◦ ı = ~ as well asσ ◦ ~ = ı. 2
Lemma 3.16 Let Z ⊆ Y ∈ WELL, ı : Y 1−1→ B and : Y 1−1→ B. Furthermore, supposethat ı ¹ Z = ¹ Z and B \ (ı[Y] ∪ [Y]) is infinite. Then, for all formulae φ(~u, ~U, W ),~x ∈ S and ~X ∈ CLY,ı satisfying (suppB(~x) ∪ suppB( ~X)) ∩ (ı[Y \ Z] ∪ [Y \ Z]) = ∅,
In particular, the latter holds if suppB(~x) ∪ suppB( ~X) ⊆ ı[Z].
Proof : Since B \ (ı[Y] ∪ [Y]) is infinite, we can find an A ⊆ B \ (ı[Y] ∪ [Y]) and abijection ∂ : (Y \ Z) → A. Now define ~ by ~¹Z := ı¹Z and ~¹(Y \ Z) := ∂.
By Lemma 3.15, we may select ρ, τ ∈ Aut(B/F ) such that ρ : MY,ı ∼= MY,~ andτ : MY, ∼= MY,~. As a result, τ−1 ◦ ρ ∈ Aut(B/F ), τ−1 ◦ ρ : MY,ı ∼= MY,, andτ−1 ◦ ρ(ı(R)) = (R). Any b ∈ F satisfies ρ(b) = b = τ(b). Hence, letting σ := τ−1 ◦ ρ,we obtain σ(b) = b for any b ∈ B with suppB(b) ⊆ F . Thus the desired assertion followsby Proposition 3.14. 2
3.4 The model M
In this subsection we introduce the structures MVn,hn and the class model M and showthat M satisfies Join and UMIDI. Here we are going to reason in KPw +Σ1-Sep unlessindicated otherwise.
Definition 3.17 Let Vn := WELL ∩ Lσn , hn := gn ¹Vn and h∞ :=⋃
n∈ω hn. Put
M :=⋃n
MVn,hn
i.e. M := 〈S,CLM
,◦∈
M,AppS,k, s,p,p0,p1,d, sN,pN,0,N, I, (cm)m∈ω, i, j, lfp〉,
where CLM
:=⋃
n CLVn,hn and◦∈
M:=
⋃n
◦∈Vn,hn
.
15
Lemma 3.18 Let ψ(~u, ~U) be an arbitrary formula and ~X ∈ CLVn0 ,hn0 . If MVn0 ,hn0 |=ψ[~x, ~X], then there exists ρ ∈ Aut(B/ran(hn0)) such that ρ = ρ−1, ρ( ~X) ∈ CLVn0 ,hn0 ,(suppB(ρ(~x)) ∪ suppB(ρ( ~X))) ∩ ran(h∞) ⊆ ran(hn0), and MVn0 ,hn0 |= ψ[ρ(~x), ρ( ~X)].
Proof : Let
G := ((suppB(~x) ∪ suppB( ~X)) ∩ ran(h∞)) \ ran(hn0).
As suppB(~x) ∪ suppB( ~X) is a finite set there exists a prime number p > 5 such that
(suppB(~x) ∪ suppB( ~X)) ∩ {pi+1 : i ∈ ω} = ∅.
Pick U ⊆ {pi+1 : i ∈ ω} such that U and G have the same number of elements, and let= : G → U be a bijection. Note that ran(hn0) ⊆ {5i+1 : i ∈ ω} according to 2.15. Wemay then define ρ ∈ Aut(B/ran(hn0)) by:
ρ(x) :=
x if x /∈ (G ∪ U)=(x) if x ∈ G=−1(x) if x ∈ U .
As a result,ρ[G] ⊆ {pi+1 : i ∈ ω} and ρ = ρ−1.
On account of MVn0 ,hn0 |= ψ[~x, ~X] and Proposition 3.14, we obtain MVn0 ,ρ◦hn0 |=ψ[ρ(~x), ρ( ~X)]. But ρ ◦ hn0 = hn0 , and hence MVn0 ,hn0 |= ψ[ρ(~x), ρ( ~X)]. 2
Corollary 3.19 Let ψ(~u, ~U) be an arbitrary formula. For meta n0 < n1 the followingis provable in KPr + Σ1-Sep: If ~X ∈ CLVn0 ,hn0 and MVn0 ,hn0 |= ψ[~x, ~X], then thereexists ρ ∈ Aut(B/ran(hn0)) such that ρ = ρ−1, ρ( ~X) ∈ CLVn0 ,hn0 , (suppB(ρ(~x)) ∪suppB(ρ( ~X))) ∩ ran(hn1) ⊆ ran(hn0), and MVn0 ,hn0 |= ψ[ρ(~x), ρ( ~X)].
Proof : Replace h∞ with hn1 in the proof of 3.18. 2
Theorem 3.20 Let φ(~u, ~U, W ) be a ΣEM-formula and n0 ≤ n1. Suppose
(suppB(~x) ∪ suppB( ~X)) ∩ ran(hn1) ⊆ ran(hn0)
and ~X ∈ CLVn0 ,hn0 . If ∃R ∈ Vn1 MVn1 ,hn1 |= φ[~x, ~X,hn1(R)], then ∃R ∈ Vn0 MVn0 ,hn0 |=φ[~x, ~X,hn0(R)].
Proof : Let
F := (suppB(~x) ∪ suppB( ~X)) ∩ ran(hn1);Z := (hn1)
−1[F ] = (hn0)−1[F ].
Then Z ∈ Lσn0and hn1 ¹Z = hn0 ¹Z ∈ Lσn0
. We may pick ` : Vn1
1−1→ B such that
`¹Z = hn1 ¹Z ∧ ` [Vn1\Z] ⊆ {pi+1 : i ∈ ω},
where p is a prime number p > 5 such that (suppB(~x) ∪ suppB( ~X)) ∩ {pi+1 : i ∈ ω} = ∅.This is possible since suppB(~x) ∪ suppB( ~X) is a finite set. We then have (suppB(~x) ∪
The latter statement is a Σ formula whose parameters are ~X, ~x,Z, and hn0 ¹ Z. Thelatter being elements of Lσn0
, (2) must already hold in Lσn0. As a result, we may choose
X and from Lσn0. We have (suppB(~x) ∪ suppB( ~X)) ∩ ( [X \Z] ∪ hn0 [X \Z]) = ∅,
thus, putting to use Lemma 3.16 again, ∃R ∈ X MX , |= φ[~x, ~X, (R)] implies
∃R ∈ X MX ,hn0¹X |= φ[~x, ~X,hn0(R)].
Recall that φ is a ΣEM-formula. Hence, by Corollary 3.12, we then get ∃R ∈ X MVn0 ,hn0 |=φ[~x, ~X,hn0(R)]. 2
Corollary 3.21 Let φ(~u, ~U,W ) be a ΣEM-formula. For meta n0 < n1 the followingis provable in KPr + Σ1-Sep: If (suppB(~x) ∪ suppB( ~X)) ∩ ran(hn1) ⊆ ran(hn0),~X ∈ CLVn0 ,hn0 and ∃R ∈ Vn1 MVn1 ,hn1 |= φ[~x, ~X,hn1(R)], then ∃R ∈ Vn0 MVn0 ,hn0 |=φ[~x, ~X,hn0(R)].
Proposition 3.22 M is a model of Join.
Proof : Suppose A ∈ CLM
and M |= ∀x ◦∈ A∃Y fx ' Y . Set
F := (suppB(A) ∪ suppB(f)) ∩ ran(h∞).
Note that F is finite. Thus, by definition of M, there exists n0 such that A ∈ CLVn0 ,hn0
and F ⊆ ran(hn0).Now suppose M |= x0
◦∈ A. Then, using 3.10,
MVn0 ,hn0 |= x0◦∈ A. (3)
By 3.18, there exists ρ ∈ Aut(B/ran(hn0)) such that ρ = ρ−1, ρ(A) = A, ρ(f) = f ,suppB(ρ(x0)) ∩ ran(h∞) ⊆ ran(hn0), and MVn0 ,hn0 |= ρ(x0)
◦∈ A. Consequently,
M |= ∃Y [fρ(x0) ' Y ].
Whence, for some n1 ≥ n0,
MVn1 ,hn1 |= ∃Y [fρ(x0) ' Y ]. (4)
Note that(suppB(ρ(x0)) ∪ suppB(f) ∪ suppB(A)
) ∩ ran(hn1) ⊆ ran(hn0).Therefore, by 3.20, (4) implies
MVn0 ,hn0 |= ∃Y [fρ(x0) ' Y ].
Since hn0 = ρ ◦ hn0 , we may deduce
MVn0 ,hn0 |= ∃Y [ρ(f)ρ(ρ(x0)) ' Y ]
17
by Proposition 3.14. As ρ(f) = f and ρ = ρ−1, the latter provides
MVn0 ,hn0 |= ∃Y [fx0 ' Y ].
As x0◦∈ A was arbitrary, we have shown
MVn0 ,hn0 |= ∀x ◦∈ A∃Y [fx ' Y ]. (5)
On account of the definition of MVn0 ,hn0 , (5) yields
MVn0 ,hn0 |= ∃X j(A, f) ' X,
and thusM |= ∃X j(A, f) ' X.
2
Lemma 3.23 If A ∈ CLVn0 ,hn0 , M |= ∀X ◦⊆ A∃Y ◦⊆ AfX ' Y and(suppB(f) ∪
suppB(A)) ∩ ran(h∞) ⊆ ran(hn0), then
MVn0 ,hn0 |= ∀X ◦⊆ A∃Y ◦⊆ A fX ' Y.
Proof : Suppose M |= ∀X ◦⊆ A∃Y ◦⊆ AfX ' Y and(suppB(f) ∪ suppB(A)
) ∩ran(h∞) ⊆ ran(hn0). Fix X0 ∈ CLVn0 ,hn0 such that MVn0 ,hn0 |= X0
◦⊆ A. By3.18 there exists ρ ∈ Aut(B/ran(hn0)) such that ρ = ρ−1, ρ(f) = f , ρ(A) = A,
As (suppB(f) ∪ suppB(ρ(X0))) ∩ ran(hn1) ⊆ ran(hn0), we obtain
MVn0 ,hn0 |= ∃Y [fρ(X0) ' Y ], (7)
utilizing (6) and 3.20. The latter implies MVn0 ,hn0 |= ∃Y [ρ(f)ρ(ρ(X0)) ' Y ] by 3.14since hn0 = ρ ◦ hn0 . But ρ(f) = f and ρ(ρ(X0)) = X0. Hence
MVn0 ,hn0 |= ∃Y [fX0 ' Y ].
As X0 ∈ CLVn0 ,hn0 was arbitrary, we get MVn0 ,hn0 |= ∀X ◦⊆ A∃Y ◦⊆ AfX ' Y. 2
Corollary 3.24 For meta n0 < n the following is provable in KPr + Σ1-Sep: If A ∈CLVn0 ,hn0 , MVn,hn |= ∀X ◦⊆ A∃Y ◦⊆ AfX ' Y and
(suppB(f) ∪ suppB(A)
) ∩ran(hn) ⊆ ran(hn0), then
MVn0 ,hn0 |= ∀X ◦⊆ A∃Y ◦⊆ A fX ' Y.
18
Proof : Replace M with MVn,hn and h∞ with hn in the proof of 3.23. 2
Proposition 3.25 M |= UMIDI.
Proof : Suppose M |= Clop(f, I) ∧ Mon(f, I). Let
F := suppB(f) ∩ ran(h∞).
Then F ⊆ ran(hn0) for some n0. Let n1 > n0. Owing to Lemma 3.23, we have
MVn1 ,hn1 |= ∀X ◦⊆ I∃Y ◦⊆ I fX ' Y. (8)
Further, (8) and M |= Mon(f, I) imply
MVn1 ,hn1 |= Clop(f, I) ∧ Mon(f, I). (9)
LettingR := {〈fn0(η), fn0(ξ)〉 : η < ξ < σn0}
(fn0 has been defined in Lemma 2.14), we obtain a well-ordering R ∈ Vn1 of ω of lengthσn0 . Using (9) and the fact that MVn1 ,hn1 |= T0, there exist A,B ∈ CLVn1 ,hn1 suchthat
MVn1 ,hn1 |= ItN(R, f) ' A ∧ fA ' B, (10)
where R := hn1(R). Next, we aim at showing
MVn1 ,hn1 |= B◦⊆ A. (11)
Suppose MVn1 ,hn1 |= x0◦∈ B. Then
∃R ∈ Vn1 MVn1 ,hn1 |= ∃Y ∃Z [ItN(τ(hn1(R)), f) ' Y ∧ fY ' Z ∧ x0◦∈ Z ].
Note that suppB(x0) = ∅ as MVn1 ,hn1 |= x0◦∈ I. Therefore, by Theorem 3.20, the above
implies
∃R∗ ∈ Vn0 MVn0 ,hn0 |= ∃Y ∃Z [ItN(τ(hn0(R∗)), f) ' Y ∧ fY ' Z ∧ x0
◦∈ Z ].
As the order-type of R∗ is strictly smaller than the order-type of R, the latter impliesMVn0 ,hn0 |= x0
◦∈ A, confirming (11). Using (11), the definition of MVn1 ,hn1 thenensures lfp(f) ∈ CLVn1 ,hn1 . As M |= Clop(f, I) ∧ Mon(f, I), lfp(f) is clearly the leastfixed-point of f in M. Therefore M |= UMIDI. 2
Corollary 3.26 For meta n0 < n1 the following is provable in KPr + Σ1-Sep: IfMVn1 ,hn1 |= Clop(f, I) ∧ Mon(f, I) and suppB(f) ∩ ran(hn1) ⊆ ran(hn0), thenlfp(f) ∈ CLVn1 ,hn1 and MVn1 ,hn1 |= Lfp(lfp(f), f, I).
Proof : Replace M with MVn1 ,hn1 and h∞ with hn1 in the proof of 3.25 and use 3.24instead of 3.23. 2
Open Problem 3.27 The proof of Proposition 3.25 lends itself to the question whetherM is actually a model of UMID. In the proof of M |= UMIDI we exploited the centralfact that the elements of I are invariant under automorphisms. However, we conjecturethat M |= UMID.
19
3.5 Some upper bounds
Theorem 3.28 (KPw + Σ1-Sep) M |= T0(I)¹ +INDN + UMIDI.(To be more precise, for every theorem ψ of T0(I)¹ +INDN + UMIDI we have
KPw + Σ1-Sep ` M |= ψ.)
Proof : Clearly, M is a model of the applicative axioms and the axioms for naturalnumbers. Furthermore, in view of Corollary 3.10 and the definition of the MVn,hn ’s, itfollows that M satisfies (ECA), INDN, and IG¹. Further, by Proposition 3.22, M is amodel of Join, and, by Proposition 3.25, M is a model of UMIDI. 2
Theorem 3.29 (KP + Σ1-Sep) M |= T0(I) + UMIDI.(To be more precise, for every theorem ψ of T0(I) + UMIDI we have KP+Σ1-Sep `
M |= ψ.)
Proof : One uses Foundation to verify M |= IG. The rest follows from 3.28.To show M |= IG in more detail, assume
M |= ∀x ◦∈ A [∀y ((y, x)◦∈ B → φ(y)) → φ(x)]. (12)
We need to verify that M |= ∀x ◦∈ i(A,B) φ(x). Let a := {z ∈ S : M |= z◦∈ A}
and r := {〈y, x〉 ∈ S × S : M |= (y, x)◦∈ B}. The definition of i(A,B) yields that
{x ∈ S : M |= x◦∈ i(A,B)} = wfp(a, r). Thus, using induction on the well-founded
part of r on a (Corollary 2.10), the assumption (12) implies ∀x ∈ wfp(a, r) M |= φ(x),and hence M |= ∀x ◦∈ i(A,B)φ(x). 2
For theories T1, T2, we use the notation T1 ≤ T2 to signify that T1 is proof-theoreticallyreducible to T2.
To obtain more detailed information than Theorem 3.30 (which will be presented insection 5) one should relativize the model constructions of this section to a given setU ⊆ ω.
Definition 3.31 The structure
MX ,`U,α = 〈S,CLX ,`
U,α,◦∈X ,`
U,α,AppS,k, s,p,p0,p1,d, sN,pN,0,N, I, (cm)m∈ω, i, j, lfp,U〉
is defined as MX ,`α in Definition 3.5 except that CLX ,`
U,0 has an additional classificationU := 14◦ satisfying
z◦∈X ,`
U,0U :⇔ z ∈ U◦.
The structures MX ,`U,α give rise to the model
MX ,`U := 〈S,CLX ,`
U ,◦∈X ,`
U ,AppS,k, s,p,p0,p1,d, sN,pN,0,N, I, (cm)m∈ω, i, j, lfp,U〉,
20
where CLX ,`U :=
⋃α CLX ,`
U,α and◦∈X ,`
U :=⋃
α
◦∈X ,`
U,α. Finally let VUn := WELL ∩ LσU
n(U)
and hUn := gU
n ¹VUn and put
MU :=⋃n
MVU
n ,hUn
U .
The results of this section all carry over to MU . As a strengthening of Theorem 3.29 weobtain:
Theorem 3.32 If T0(I) + UMIDI ` ψ, then KP + Σ1-Sep ` ∀U ⊆ ω MU |= ψ.
4 Reducing T0(I)¹ +UMIDI to KPr + Σ1-Sep
Since the model M cannot be formalized in KPr + Σ1-Sep we cannot use it to reduceT0(I) ¹ +UMIDI to KPr + Σ1-Sep. However, for meta n, the existence of MVn,hn
can already be proved in KPr + Σ1-Sep. Settling for a partial interpretation, we shallshow that those theorems of T0 ¹ +UMIDN, wherein all the classification quantifiers areexistential, hold in any of the models MVn,hn . The latter reduces T0(I) ¹ +UMIDI toKPr + Σ1-Sep. Technically, the first step consists in proving a partial cut-eliminationresult for T0(I)¹ +UMIDI.
4.1 A Tait-style calculus for explicit mathematics
The Tait-style calculus to be developed in this subsection relies on a slightly differentaccount of the language of explicit mathematics. Namely, the Tait language LT onlycontains the logical symbols ∧, ∨, ∀, ∃, but has the additional relation symbols 6=,
∼App,◦/∈ to express the complements of =, App,
◦∈, respectively. Negation then becomesa defined operation on the formulae in the obvious way, using the de Morgan laws topush it through to the prime formulae.
Definition 4.1 The ΣEM-formulae form the least class of formulae containing the quanti-fier-free formulae which is closed under ∧, ∨, object quantification, and ∃-quantificationover classifications.
The ΠEM-formulae form the least class of formulae containing the quantifier-freeformulae which is closed under ∧, ∨, object quantification, and ∀-quantification overclassifications.
∆EM-formulae of LT are formulae which are both ΣEM- and ΠEM-formulae, i.e. for-mulae which do not contain any unbounded classification quantifiers.
ΣEM1 -formulae are formulae of the form ∃X1 . . .∃Xkφ(X1, . . . , Xk) where φ does not
contain classification quantifiers. ΠEM1 -formulae are the negations of ΣEM
1 -formulae.
The idea now is to embed theories from explicit mathematics into the Tait-calculusand then to perform a partial cut-elimination which only leaves us with cuts on ΣEM
1 -and ΠEM
1 -formulae. For this to work we have to make some minor adjustments. First,we need an adequate definition of the rank of a formula.
Definition 4.2 The rank of an LT -formula is the rank over its ΣEM1 - and ΠEM
1 -subformulae.Formally, if φ is a ΣEM
1 - or ΠEM1 -formula, then rk(φ) = 0. If φ is not of the latter form,
the rank is defined as follows:
21
a) If φ is of the form φ0 ∧ φ1 or φ0 ∨ φ1, then rk(φ) = max{rk(φ0), rk(φ1)}+ 1.b) If φ is of the form ∃xψ(x), ∀xψ(x), ∃Xψ(X), or ∀Xψ(X), then rk(φ) = rk(ψ) + 1.
The second adjustment is to make sure that all formulae introduced by non-logicalaxioms and rules are ΣEM
1 . For this it is convenient to switch to a slightly differentformulation of the join axiom which has a syntactically simpler form.
Lemma 4.3 The applicative fragment of EM0 ¹ proves that under the hypothesis ∀x ◦∈A∃X(fx ' X) the following assertions are equivalent:
(i) ∃Z Join(f, A, Z), i.e. ∃Z(Z ' j(f, A)∧∀z(z◦∈ Z ↔ ∃x ◦∈ A∃y(z ' (x, y)∧y
◦∈ fx)).
(ii) ∀z∃Z Joinv(f, z, A, Z) where
Joinv(f, z, A, Z) ≡ ∃Y ∃X(Z ' j(f, a) ∧(z
◦∈ Z → p0z◦∈ A ∧ Y ' f(p0z) ∧ p1z
◦∈ Y ) ∧(p0z
◦∈ A ∧ (X ' f(p0z) → p1z◦∈ X) → z
◦∈ Z)).
Proof : Argue in the applicative fragment of EM0 ¹. If ∀x ◦∈ A (∃X(fx ' X)), thenthese X are unique. Therefore
∃Z Join(f, A, Z) ⇔ ∀z∃Z(Z ' j(f,A) ∧ (z◦∈ Z ↔ ∃x ◦∈ A∃y(z ' (x, y) ∧ y
◦∈ fx))⇔ ∀z∃Z [
Z ' j(f,A) ∧(z
◦∈ Z → p0z◦∈ A ∧ ∃Y (Y ' f(p0z) ∧ p1z
◦∈ Y )) ∧(p0z
◦∈ A ∧ ∀X(X ' f(p0z) → p1z◦∈ X) → z
◦∈ Z)]
⇔ ∀z∃Z Joinv(f, z, A, Z)
2
Definition 4.4 The calculus T is defined as follows:
a) Logical axioms(Ax) Γ,¬φ, φ where rk(φ) = 0.
b) Equality axioms
(Eq1) Γ, t = t for object terms t.
(Eq2) Γ, s 6= t,¬φ(s), φ(t) where rk(φ) = 0.
c) Logical rules
(∧)Γ, φ0 Γ, φ1
Γ, φ0 ∧ φ1(∨)
Γ, φi
Γ, φ0 ∨ φ1i = 0, 1
(∀0)Γ, φ(a)
Γ,∀xφ(x)(∃0)
Γ, φ(t)Γ, ∃xφ(x)
(∀1)Γ, φ(A)
Γ, ∀Xφ(X)(∃1)
Γ, φ(A)Γ, ∃Xφ(X)
As usual we have the proviso that the free variables a and A in (∀0) and (∀1),respectively, are not to occur in the respective conclusion.
22
d) Non-logical axiomsΓ, φ
where φ is one of the following:
• an instance of an applicative axiom.
• an instance of (ECA), i.e. ∃X(X ' cm(~t, ~A) ∧ ∀x(x◦∈ X ↔ F (x,~t, ~A))) for
certain terms ~t and classification variables and constants ~A.
• the induction axiom
0◦∈ A ∧ ∀x ◦∈ N(x
◦∈ A → sNx◦∈ A) → ∀x ◦∈ N(x
◦∈ A).
• the open form of (IG)¹, which is separated into two axioms,
kΓ to convey that there exists a derivation in T in which
all cut-formulae have rank less than k and which is of length ≤ n.The definition of the calculus T is tailored so that the following proposition holds:
Proposition 4.5 If EM0 ¹ +(IG)¹ +(J) ` φ, then there are n, k < ω such that T n
kφ.
Proof : This is standard. 2
Since all non-logical axioms and rules only introduce formulae of rank 0, we caneliminate all cuts of higher complexity from our derivations. In other words:
Proposition 4.6 If T m
kΓ, then there is some n such that T n
1Γ. In point of fact, one
has n ≤ 2k−1(m) where 20(m) = m and 2r+1(m) = 22r(m).
Proof : Standard cut-elimination. 2
When we combine the previous propositions, we obtain:
Proposition 4.7 If EM0 ¹ +(IG) ¹ +(J) ` φ, then there is some n < ω such thatT n
1φ.
To treat UMIDI in this context we again (as in the case of (J)) prefer to use a slightvariant of the axiom which is in a syntactic form that can be dealt with in an easier wayin the following.
Lemma 4.8 The applicative fragment of EM0 ¹ proves: If Clop(f , I), then the followingformulations of the least fixed-point axiom are equivalent.
where the official renderings of Clop(f ,A) and Mon(f, A) are as follows:
Clop(f ,A) := ∀X ∃Y [X◦⊆ A → Y
◦⊆ A ∧ fX ' Y ]
Mon(f, A) := ∀X ∀Y [X◦⊆ A ∧ Y
◦⊆ A ∧ X◦⊆ Y → fX
◦⊆ fY ]
with fX◦⊆ fY being a shorthand for ∃Z∃W (fX ' Z ∧ fY ' W ∧ Z
◦⊆ W ).
Proof : Similar to Lemma 4.3. 2
4.2 The interpretation
In the following we shall use variables A,B, C, . . . to range over classifications of struc-tures of the form MX ,`, i.e. CLX ,`. a, b, c, . . . will be ranging over elements of S. Theseconventions are necessitated by our desire to differentiate between the variables of theformal theory T0(I)¹ +UmidI and variables ranging over the realms over their interpre-tations.
Proposition 4.9 Let Γ[~a, ~A] be a set of ΣEM-formulae. If T n
1¬(UmidI), Γ[~a, ~A], then,
for all (meta) m, the theory KPr + Σ1-Sep proves:
∀ ~A ∈ CLVm,hm ∀~a ∈ S (MVm,hm |= Γ[~a, ~A]).
Here |= Γ[~a, ~A] is short for |= ∨Γ[~a, ~A], where
∨Γ[~a, ~A] is the disjunction over all
formulae of Γ[~a, ~A].
Proof : We will use the abbreviations
CLm := CLVm,hm , Mm := MVm,hm .
The proof proceeds by (meta) induction on n. We shall restrict our attention to the mostimportant cases, as the remaining ones easily follow using the induction hypotheses.
If Γ is an axiom, then we discern two subcases.In the first one, Γ is a ∆EM-formula (i.e. in the cases of (Ax), (Eq), applicative
axioms, induction axiom and (IG2)¹). Then the assertion holds by construction ofSM,γ . In the case of the induction axiom we have to note that for each A ∈ CLm theset {a ∈ S : Mm |= a
◦∈ A} is in Lσm and therefore we can use induction in Lσm (on theset {n◦ : n ∈ ω}) to prove the instance of the induction axiom.
In the second axiom case we have an instance of (ECA) or one of (IG1) in its openformulation. For example, let us treat (IG1). For arbitrary m and A0,A1 ∈ CLm wehave i(A0,A1) ∈ CLm and so the assertion is established.
24
We leave out the propositional, quantifier and equality rules, since they can be treatedusing the induction hypotheses. But note that it is important that there are no (∀1)-rulesto be considered as fact that Γ consists of ΣEM-formulae solely.
Case 2: The last inference was a cut. As the cut formulae possess rank 0, we then havepremises of the shape
T n0
1Γ[~a, ~A],∃~Y φ[~a,~b, ~Y , ~A, ~B]
andT n1
1Γ[~a, ~A],∀~Y ¬φ[~a,~b, ~Y , ~A, ~B],
where φ is a ∆EM-formula and n0, n1 < n. Application of the induction hypothesis tothe first premise yields
for all m ∈ N and ~a ∈ S, where ~∅ is a tuple of ∅’s and ~0 is a tuple of 0’s of appropriatelengths with ∅ being the empty classification. Applying (∀1) inversions to the secondpremise we get
T n1
1Γ[~a, ~A],¬φ[~a,~b, ~C, ~A, ~B]
for new classification variables ~C. Applying the induction hypothesis to this derivationwe get
for all ~a ∈ S.Now assume that there are ~a ∈ S, ~A ∈ CLm such that Mm 6|= Γ[~a, ~A]. Then (13)
supplies us with ~B ∈ CLm such that
Mm |= φ[~a, ~0, ~B, ~A,~∅]. (15)
Using (14), we get
Mm |= Γ[~a, ~A],¬φ[~a,~0, ~B, ~A,~∅]. (16)
From (15) and (16) we deduceMm |= Γ[~a, ~A]
contradicting our assumption. Thus Mm |= Γ[~a, ~A] must be true.
Case 3: The last inference is (JR). Then a formula ∃Z Joinv(t, s, Ai, Z) is in Γ and wehave
T n0
1Γ[~a, ~A],∀y ◦∈ Ai∃Y (ty ' Y )
for some n0 < n. Fix ~A ∈ CLm and ~a ∈ S and let f := t[~a, ~A]. Assume Mm 6|=Γ[~a, ~A]. The induction hypothesis then yields Mm |= ∀b ◦∈ Ai∃B(fb ' B) and thereforej(f,Ai) ∈ CLm by the definition of CLVm,hm . Consequently, Mm |= ∃C Join(f,Ai, C)and therefore Mm |= ∀c∃C Joinv(f, c,A, C). But the latter implies Mm |= Γ[~a, ~A],colliding with our assumption. Therefore Mm |= Γ[~a, ~A] holds.
Case 4: Assume now (this is the pivotal case) that the last inference was an (∃0)-inference with main formula ¬(UmidI). Then we have a scenario of the form
for n0 < n and a term t whose free variables are among ~a,~b, ~A, ~B. Using (∧) inversionsfollowed by a (∀1) inversion, we get the following derivations
where B,C are new free classification variables.Now assume m ∈ N, ~A ∈ CLm and ~a ∈ S. Set f := t[~a, ~0, ~A,~∅].Define k := m + 1. By 3.19 there exists ρ ∈ Aut(B/ran(hm)) such that ρ = ρ−1,
(⋃
i
suppB(ρ(ai)) ∪⋃
j
suppB(ρ(Aj)) ∪ suppB(ρ(f))) ∩ ran(hk) ⊆ ran(hm)
and ρ( ~A) ∈ CLm.We now distinguish two subcases.
Subcase 4.1: Assume Mk |= Γ[ρ(~a), ρ( ~A)]. Employing Corollary 3.21, we thenobtain Mm |= Γ[ρ(~a), ρ( ~A)]. The latter yields Mm |= Γ[ρ(ρ(~a)), ρ(ρ( ~A))] due to 3.14;thence Mm |= Γ[~a, ~A] as ρ = ρ−1.
Subcase 4.2: Assume Mk 6|= Γ[ρ(~a), ρ( ~A)]. Set f := ρ(f). Substituting ρ(~a), ρ( ~A)for ~a, ~A, the induction hypotheses pertaining to (18) and (17) supply
∀B ∈ CLk Mk |= ∃Y (B ◦⊆ I → Y◦⊆ I ∧ fB ' Y ), (20)
∀B, C ∈ CLk Mk |= B ◦⊆ I ∧ C ◦⊆ I ∧ B ◦⊆ C → fB ◦⊆ fC, (21)∀B ∈ CLk Mk |= lfp(f) ' B → ¬Lfpv(B, f , I). (22)
Recall that suppB(f) ∩ ran(hk) ⊆ ran(hm). (20) and (21) yield Mk |= Clop(f , I) ∧Mon(f , I). Thus, putting to use Corollary 3.26, we obtain lfp(f) ∈ CLVk,hk and Mk |=Lfp(lfp(f), f , I). The latter yields Mk |= Lfpv(lfp(f), f , I) which collides with (22).Consequently, this subcase cannot occur and we are back to the first subcase. As aresult we have Mm |= Γ[~a, ~A] as desired. 2
Corollary 4.10 If T0(I) ¹ +UMIDI ` φ(~x, ~X) for a ΣEM-formula φ, then, for all m,KPr + Σ1-Sep proves that for all ~a ∈ S and ~A ∈ CLV m,hm, MVm,hm |= φ[~a, ~A].
Theorem 4.11 T0(I)¹ +UMIDI ≤ KPr + Σ1-Sep.
4.3 Relativizations
The results of this section can be relativized to the models MVU
n ,hUn
U of Definition 3.31.In particular we get:
Corollary 4.12 If T0(I) ¹ +UMIDI ` φ(~x, ~X) for a ΣEM-formula φ, then, for all m,KPr + Σ1-Sep proves that for all U ⊆ ω, ~a ∈ S and ~A ∈ CLVU
m,hUm, M
VUm,hU
mU |= φ[~a, ~A].
26
5 Reducing set theory to analysis and vice versa
In this section the circle of reductions is completed by showing that KPr + Σ1-Sep andKPw + Σ1-Sep prove the same theorems of second order arithmetic as (Π1
2 − CA) ¹and (Π1
2 − CA), respectively. The language of second-order arithmetic is regarded asa sublanguage of set theory via the translation mapping numerical quantifiers ∃x to∃x(x ∈ ω ∧ . . . ) and set quantifiers ∃X to ∃X(X ⊆ ω ∧ . . . ). We also show thatKP + Σ1-Separation is a conservative extension of (Π1
2 − CA) + BI, where BI is theso-called principle of Bar Induction, i.e. the axiom schema
∀X (WO(<X) ∧ ∀n [∀m <X nΦ(m) → Φ(n)
] → ∀nΦ(n))
for all formulae Φ of the language of second order arithmetic, where m <X n := 2m ·3n ∈X.
Definition 5.1 a) A non-empty transitive set A is called an admissible set if
〈A,∈〉 |= KP.
b) An ordinal α is called admissible, if Lα is an admissible set.
Recall that Infinity is assumed to be among the axioms of KP. The next lemma isneeded for proving Π1
2 Comprehension in KPr + Σ1-Sep.
Lemma 5.2 KPr + Σ1-Sep ` ∀x ⊆ ω ∃y [x∈ y ∧ y is an admissible set ].
Proof : Suppose X ⊆ ω. Let L(X) be the class of all sets constructible from X. Notethat L(X) is naturally equipped with a ∆1 definable well-ordering <L(X) since X inheritsa well-ordering from ω.
Let AX be the set of those a∈L(X) for which there is a Σ1 definition of a in L(X)using the parameter X. To be more precise, let
BX = {pφ(u, v, w)q : pφ(u, v, w)q is the Godel number of a ∆0 (23)formula φ s.t. L(X) ² ∃y∃zφ(y, z,X)}
Using a Σ1 satisfaction predicate, one sees that BX is Σ1 definable and thus BX is a setby Σ1 Separation. Then ran(F ) is a set by Σ collection and consequently AX is a set.Obviously
〈AX , ∈ ∩ A×A〉 ≺1 L(X). (26)
Now let cAXbe the Mostowski collapsing function on AX . Then ran(cAX
) is an admis-sible set due to (26), and, in addition, this set contains X since
cAX(X) = {cAX
(n) : n∈X} = {n : n∈X} = X.
This proves our lemma. 2
27
Theorem 5.3 KPr + Σ1-Sep and (Π12 − CA) ¹ prove the same sentences of second
order arithmetic.
Proof : “⊇”: We shall be arguing informally in KPr + Σ1-Sep.First, we address Comprehension. Instead of Π1
2 Comprehension we may as well showΣ1
2 Comprehension.In the set-theoretic language a Σ1
2 formula becomes a formula
∃x ⊆ ω ∀y ⊆ ω ψ(n, x, y)
where ψ is arithmetic, i.e. all quantifiers in ψ are bounded by ω. Now with any Π11 formula
θ(u,U) with free variables u and U ranging over ω and subsets of ω, respectively, onecan associate an arithmetic formula y ≺u,U z such that for all X ⊆ ω, ≺n,X is a binaryrelation on ω and given n∈ω,
θ(n,X) iff ≺n,X is well-founded. (27)
For any binary relation ≺ on ω we define an operation C≺ via Σ Recursion on theordinals:
Suppose now that A is an admissible set, that ≺ is well-founded and ≺ is an elementof A. Let F≺ be the restriction of C≺ to the ordinals of A. Then F≺ is a set which is Σ1
definable on A. We claim that
∀n∈ω∃α∈A [n∈F≺(α)]. (30)
If this were not the case, we would let n0 be a ≺-least integer n such thatn /∈ ⋃
ran(F≺). Consequently,
∀m ≺ n0∃α ∈ A[m∈F≺(α)].
But then, by Σ Reflection in A, there would exist α0 ∈A such that ∀m ≺ n0 [m∈F≺(α0)],yielding the contradiction n0 ∈F≺(α0 + 1).
¿From (30), using Σ Reflection in A, we obtain an α∈A such that ω ⊆ F≺(α). Thuswe obtain a function H ∈A with H : ω −→ ON by letting
H(n) = least α. n∈F≺(α). (31)
The important property that H satisfies is
∀n∀m[n ≺ m → H(n) < H(m)]. (32)
On the other hand, (32) always implies that ≺ is well-founded. So the upshot is that,in view of Lemma 5.2, the well-foundedness of a relation ≺ on ω is equivalent to the
28
existence of an admissible set A which contains ≺ and a function H ∈A satisfying (32).Let φ(H,≺) be a shorthand for (32). By the preceding, the right hand side of (29) givesthe same class as
rendering {n∈ω : ∃x ⊆ ω ∀y ⊆ ω ψ(n, x, y)} a Σ1 class and therefore a set via ΣSeparation.
“⊆”: In the course of the proof we employ the method of trees which has been used byseveral people (see [2], Sec. 5). Within (Π1
2 −CA)¹ we make the following definitions:
A tree is a non–empty set T of (codes for) finite sequences of natural numbers such thats ⊆ t ∧ t ∈ T → s ∈ T . A tree T is said to be well founded if there is no function fsuch that ∀nf [n]∈T , where f [n] = 〈f(0), · · · , f(n − 1)〉. Trees T and T ′ are said to beisomorphic, written T ∼= T ′, if there exists an isomorphism between them, i.e. an orderpreserving bijection of T onto T ′. If s and t are finite sequences of natural numbers,s ? t denotes the concatenation of s followed by t. If T is a tree and s ∈ T , we writeTs = {t : s ? t∈T}.
A tree T is said to be suitable, written ST(T ), if it is well founded and, for all s∈T ,if s ? 〈m〉 ∈T and s ? 〈n〉 ∈T and Ts?〈m〉 ∼= Ts?〈n〉, then m = n.
Clearly the predicate ST is Π11. The point of the definition is that if T and T ′ are
suitable then there is at most one order preserving bijection of T onto T ′. For suitabletrees T and T ′ we write T ∈T ′ to mean ∃n [〈n〉 ∈T ′ ∧ T ∼= T ′〈n〉
]. The relations ∼= and
∈ are Σ11 on ST.
The idea is now to identify a suitable tree T with the inductively defined set
|T | = {|T〈n〉 | : 〈n〉 ∈T}
and in this way to model hereditarily countable sets within second order arithmetic (cf.[2], Sect.5, [12], [15]. The nice thing about suitable trees is that we have
|T |= |T ′ | iff T ∼= T ′, and |T |∈|T ′ | iff T ∈T ′.
Specifically, if T, T ′ are suitable trees and, for all S, S∈T iff S∈T ′, then T ∼= T ′.Now let A = 〈M,X , ...,∈〉 be a model of (Π1
2 − CA) ¹, where M = (M, · · · ) is amodel of the first order theorems of (Π1
2 − CA) ¹ and X is a subset of the power setof M . To be precise, this notation means that in A the set quantifiers range over theelements of X .
Let B ={T ∈X : A |= ST(T )
}. For T, T ′∈B set [T ] = {S∈B : A |= S ∼= T} and
[T ]∈B [T ′] iff A |= T ∈T ′.
Let B be the structure⟨{[T ] : T ∈B},∈B
⟩for the language LST .
By the above considerations, we know that B |= Extensionality. We intend to showthat B is a model of KPr + Σ1-Sep. The set-theoretic language can be interpreted intothe language of second order arithmetic as follows. Set theoretic variables are interpretedas ranging over suitable trees. The equality relation = between set theoretic variables isinterpreted as ∼=, and ∈ is interpreted as ∈.
29
For a set theoretic formula ϕ let ϕA be the corresponding second order arithmeticformula. We then get for T1, . . . , Tk∈B that
B |= ϕ([T1], . . . , [Tk]
)iff A |= ϕA(T1, . . . , Tk). (34)
Note that if ϕ happens to be a ∆0 formula, then ϕA will be equivalent to a ∆12 formula
within the structure A, because any universal bounded quantifier in ϕ gets translatedinto a quantifier of the form
∀S [ST(S) ∧ S∈T → . . .S . . .
],
where T is a suitable tree. The latter is equivalent to
∀S [∃nS ∼= T〈n〉 → . . . S . . .],
and thus equivalent to∀n[
. . . T〈n〉 . . .],
employing Extensionality.
First, we want to very that B is a model of ∆0 Collection. Suppose
B |= ∀x∈ [T ]∃yϕ(x, y)
where ψ is ∆0. For convenience, let us assume that ψ has no free variables other thanx, y. By (34) it follows
A |= ∀S∃R[ST(S) ∧ S ∈T → ST(R) ∧ ψA(S, R)
);
hence A |= ∀n∃RΘ(n,R, T ), where
Θ(S,R, T ) stands for 〈n〉∈T → ST(R) ∧ ψA(T〈n〉, R).
Now ∆12−CA proves the Σ1
2 Axiom of Choice, (Σ12−AC), since the proof of the Kondo–
Addison uniformization theorem can be done within ∆12 −CA. Θ(S,R, T ) is equivalent
to a Σ12 formula. So, by Σ1
2 −AC in A, there is an X∈X such that
A |= ∀nΘ(n, (X)n, T ). (35)
Setting
V ={t∈M : A |= ∃n∃s(t = 〈n〉 ? s ∧ 〈n〉∈T ∧ s∈(X)n ∧ ∀m < n [(X)m 6∼= (X)n]
)},
we have V ∈X by ∆12 −CA in A. By the very definition of V it follows
A |= ST(V ) ∧ ∀n∃R ∈V Θ(n,R, T ), (36)
thus B |= ∀x∈ [T ]∃y∈ [V ]ψ(x, y). This verifies B |= ∆0 Collection.
Next we verify Σ1 Separation. Consider a Σ1 class in B:
K = {[S] : B |= [S] ∈ [T ] ∧ ∃yψ([S], y, [T ], [P ])} (37)
30
with ψ being ∆0. Then
K = {[T〈m〉] : 〈m〉 ∈T ∧ B |= ∃yψ([T〈m〉], y, [T ], [P ])}. (38)
Put
Y = {m : 〈m〉 ∈T ; A |= ∃R[ST(R) ∧ ψA(T〈m〉, R, T, P )
]. (39)
Since the defining formula can be rendered Σ12, we have Y ∈M. Now define
T ∗ = {〈m〉 ? s : 〈m〉 ? s∈ T ∧ m ∈ Y }. (40)
Then T ∗ is a suitable tree in A and
B |= ∀x[x ∈ [T∗] ↔ x ∈ [T] ∧ ∃yψ(x, y, [T], [P]). (41)
This shows that B is a model of Σ1 Separation.
The verification of the remaining axioms of KPr + Σ1-Sep is routine.
In the rest of the proof we are going to show that the second order arithmetic part of B,that is to say ⟨
ωB, Pow(ω)B,∈B ∩ (ωB × Pow(ω)B)⟩
is isomorphic to A, so that the same sentences of second order arithmetic hold true in A
Then S ∈Tn iff S ∼= (Tn)〈m〉 = Tm for some m < n, and Tn ∈TX iff n∈X.The mapping i : A → (
ωB,Pow(ω)B)
determined by n 7→ [Tn] and X 7→ [TX ]then provides the desired isomorphism. Therefore the same sentences of second orderarithmetic hold in A and B.
It should be clear how the model B lends itself to a syntactic translation of KPr +Σ1-Sep into (Π1
2 −CA)¹. When in doubt, consult [12], Sec. 7. 2
Theorem 5.4 The following theories prove the same sentences of second order arith-metic:
(i) KPw + Σ1-Sep and (Π12 −CA).
(ii) KP + Σ1-Sep and (Π12 −CA) + BI.
Proof : In the proof we shall refer to definitions made in the proof of 5.3.(i): It is obvious that KPw + Σ1-Sep proves the (translation) of IND. Thus, in
view of Theorem 5.3 and its proof, it remains to verify that B is a model of the schemaof induction over ω. Well, the role of ω in B is played by
Tω :={〈n〉 ? s : s∈Tn ; n is a natural number
},
where Tn is defined as in (42). Consequently, the schema of induction over ω in B followsfrom the induction schema IND in A.
31
(ii): “⊇”: By Theorem 5.3, it remains to verify Bar Induction. Let ≺ be a well-foundedrelation on ω. We have to verify transfinite induction along ≺ for arbitrary classes inour background theory.
To this end, we define an operation C≺ via Σ Recursion on the ordinals:
C≺(α) = {n∈ω : ∀m[m ≺ n → m∈⋃
β<α
C≺(β)]}. (44)
Employing Σ Separation,
X≺ = {n∈ω : ∃α [n∈C≺(α)]} (45)
is a set. We claim that
X≺ = ω. (46)
If this were not the case, let n0 be a ≺-least integer such that n0 /∈ X≺. This implies ∀m ≺n0 [m∈X≺], and thus ∀m ≺ n0∃α∈A [m∈C≺(α)]. But then, by Σ Reflection, therewould exist α0 such that ∀m ≺ n0 [m∈C≺(α0)], yielding the contradiction n0 ∈C≺(α) ⊆X≺.
By virtue of (46), we obtain a function G : ω −→ ON by letting
G(n) = least α. n∈C≺(α). (47)
Since G satisfies ∀n∀m[n ≺ m → G(n) < G(m)], transfinite induction along ≺ forarbitrary classes follows from induction over ordinals, i.e. Foundation.
“⊆”: In view of Theorem 5.3 it remains to verify that B is a model of Foundation. Letϕ be ∆0. Assume
B |= ∀x[∀y∈xϕ(y) → ϕ(x)].
Then, for [T ]∈B, we must prove B |= ϕ([T ]). By (34) we have
A |= ∀S[ST(S) ∧ ∀n(〈n〉 ∈S → ϕA(S〈n〉)
) → ϕA(S)]. (48)
Let s ≺ t iff s, t∈T and A |= s = t ? 〈m〉 for some m∈M . Then A |= WF(≺).Now (48) implies
A |= ∀t∈T[(∀s≺ t)ϕA(Ts) → ϕA(Tt)
].
By Bar Induction in A, this gives A |= ϕA(T〈〉), hence B |= ϕ([T ]). 2
6 Conclusion
Recall that for theories T1, T2, we use the notation T1 ≤ T2 to signify that T1 is proof-theoretically reducible to T2. T1 < T2 signifies that T2 is proof-theoretically strongerthan T1. T1 ≡ T2 stands for proof-theoretic equivalence.
Due to Theorem 3.30, Theorem 4.11, Theorem 5.3, Theorem 5.4 and Theorem 1.1,we obtain:
In point of fact, we obtain more specific results than the previous theorem. Recall thatany sentence φ of second order arithmetic has a canonical translation φ∗ in the languageof T0 (see [14], Definition 5.1).
Theorem 6.2 Let φ be a Π13 sentence of second order arithmetic.
Open problems The main problem left open in this paper is whether the equivalencesof Theorem 6.2 also hold with UMID in place of UMIDI. To my knowledge, the onlyconstruction of a model of T0 + UMID was given by Takahashi in Appendix 2 of [16].Unfortunately, there is just a sketch of a model of T0 + UMID in [16]. I must confessthat I never understood that construction. I conjecture though, that the structure M
of subsection 3.4 (or a slight variant of it) is already a model of UMID. A proof ofM |= UMID seems to require more “fine structure theory” of M than is provided in thepresent paper. To show that M |= UMIDI we utilized the fact that the elements of I areinvariant under automorphisms. This part of the proof wouldn’t work for UMID. Butsimilar problems emerged in the construction of models for MID in [10, 16]. A fruitfulavenue to pursue might be to combine the techniques of [10] with the ones in the presentpaper.
Other interesting problems for further investigations in this area are raised in thelast section (“Outlook”) of [14]. To determine the strength of MID and UMID onthe basis of intuitionistic explicit mathematics are the most challenging among them.Here it might be useful to address first the related problem of fixed points for monotoneinductive definitions in constructive Zermelo-Fraenkel set theory (cf. [1]).
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[11] P.G. Hinman: Recursion-theoretic hierarchies (Springer, Berlin, 1978).[12] Jager, G.: Theories for admissible sets: a unifying approach to proof theory (Bibliopolis,
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