Replacement Versus Collection and Related Topics in Constructive Zermelo-Fraenkel Set Theory

Replacement versus Collection and related topics in Constructive Zermelo-Fraenkel Set Theory

Michael Rathjen∗ Department of Mathematics, Ohio State University Columbus, OH 43210, U.S.A. [email protected]

May 29, 2005

Abstract While it is known that intuitionistic ZF set theory formulated with Replacement, IZFR, does not prove Collection it is a longstanding open problem whether IZFR and intuitionistic set theory ZF formulated with Collection, IZF, have the same proof-theoretic strength. It has been conjectured that IZF proves the consistency of IZFR. This paper addresses similar questions but in respect of constructive Zermelo-Fraenkel set theory, CZF. It is shown that in the latter context the proof-theoretic strength of Replacement is the same as that of Strong Collection and also that the functional version of the Regular Extension Axiom is as strong as its relational version. Moreover, it is proved that, contrary to IZF, the strength of CZF increases if one adds an axiom asserting that the trichotomous ordinals form a set. Unlike IZF, constructive Zermelo-Fraenkel set theory is amenable to ordinal analysis and the proofs in this paper make pivotal use thereof in the guise of well-ordering proofs for ordinal representation systems.

MSC: 03F50, 03F35 Keywords: Constructive set theory, replacement, strong collection, proof-theoretic strength, trichotomous ordinals

1 Introduction

Zermelo-Fraenkel set theory, ZF, is formulated with Replacement rather than Collection, but it is readily seen that Collection is deducible in ZF. The proof, though, makes essential use of classical logic. On the other hand, it was shown in [12] that intuitionistic ZF set theory formulated

∗This material is based upon work supported by the National Science Foundation under Award No. DMS-0301162.

1 with Replacement, IZFR, does not prove Collection. While IZFR has the existence property, in intuitionistic ZF set theory based on Collection, IZF, the existence property fails for some formulas of the form

∃v [∃y θ → (∃y ∈ v)θ] (1)

(where v must not occur free in θ). All formulas of the form (1) are actually deducible in IZF, employing a combination of full Separation and Collection. IZFR and IZF not only differ with respect to some metamath- ematical property. [12] also established that they have different stocks of provably recursive functions. The provably recursive functions of IZFR are the same as those of a classical set theory T that is much weaker than ZF while IZF and ZF have the same provably recursive functions. It is a longstanding open problem, though, whether IZFR and IZF have the same 0 proof-theoretic strength or prove the same Π1-sentences of arithmetic. It has been conjectured by H. Friedman that IZFR is weaker than IZF (see [12]). This paper, however, is concerned with the strength of Replacement versus Strong Collection for constructive Zermelo-Fraenkel set theories. In contrast to IZF, systems of constructive Zermelo-Fraenkel set theory are amenable to methods from ordinal-theoretic proof theory. This work will make pivotal use of an ordinal analysis that Jägerand Pohlers gave in [14] 1 for the subsystem of second order arithmetic based on ∆2-comprehension and bar induction, BI. Here BI is the schema expressing that transfinite induction along any well-founded set relation holds for arbitrary classes; this being the pendant of set induction in set theory. A set theory which proves the same theorems of second order arithmetic is the system KPi which is an extension of Kripke-Platek set theory via an axiom that asserts the existence of many admissible sets, namely that every set is contained in an admissible set. To be able to explain the results of this paper we will be needing some definitions.

Deﬁnition 1.1 Let CZF be the acronym for Constructive Zermelo-Fraenkel set theory (see [1, 4]). Let CZF0 denote the system CZF without Subset 0 Collection. CZFR results from CZF by deleting Subset Collection and replacing Strong Collection with Replacement. Let CZFR,E be obtained from CZF by replacing Strong Collection with Replacement and Subset Collec- tion with Exponentiation, respectively. Note that Strong Collection implies Replacement and that Subset Collec- 0 tion implies Exponentiation. Thus, both CZFR and CZFR,E are subtheories of CZF. Another important axiom for constructive set theory is the so-called regular extension axiom, REA (see [3, 24]). A weaker version of REA is the functional regular extension axiom, fREA, which is formulated in terms of functions in place of multi-valued functions.

2 The main results can than be stated as follows.

0 Theorem 1.2 1. CZFR and CZF have the same proof-theoretic strength and the same provably recursive functions.

0 2. CZFR+fREA and CZF+REA have the same proof-theoretic strength and the same provably recursive functions.

The foregoing result involves the notion of proof-theoretic strength. So perhaps some words of clariﬁcation are in order. All theories T considered in the following are assumed to contain a modicum of arithmetic. For deﬁnite- ness let this mean that the system PRA of Primitive Recursive Arithmetic is contained in T , either directly or by translation (as in the case of the set theories considered in this paper).

Definition 1.3 Let T1, T2 be a pair of theories with languages L1 and L2, respectively, and let Φ be a (primitive recursive) collection of formu- lae common to both languages. Furthermore, Φ should contain the closed equations of the language of PRA. Let ProofT (x, y) be a primitive recursive proof predicate for T , i.e., ProofT (n, k) holds true iff n is the Gödelnumber of a proof of a formula φ in T with Gödelnumber k. We then say that T1 is proof-theoretically Φ-reducible to T2, written T1 ≤Φ T2, if there exists a primitive recursive function f such that

PRA ` ∀φ ∈ Φ ∀x [ProofT1 (x, φ) → ProofT2 (f(x), φ)]. (2)

T1 and T2 are said to be proof-theoretically Φ-equivalent, written T1 ≡Φ T2, if T1 ≤Φ T2 and T2 ≤Φ T1. The appropriate class Φ is revealed in the process of reduction itself, so that in the statement of theorems we simply say that T1 is proof-theoretically reducible to T2 (written T1 ≤ T2) and T1 and T2 are proof-theoretically equivalent (written T1 ≡ T2), respectively. Alternatively, we shall say that T1 and T2 have the same proof-theoretic strength when T1 ≡ T2. T1 and T2 are said to be equiconsistent iﬀ T1 ≡CE T2, where CE stands for the collection of closed equations of PRA. In all of the proof-theoretic reductions mentioned or established in this 0 paper, the class Φ will actually comprise the Π2 sentences of PRA. For more details about proof-theoretic ordinals of theories and the way how ordinal analyses of theories are used to establish proof-theoretic reductions see [22].

Another topic addressed in this paper is an “exotic” principle that says that the trichotomous ordinals form a set. An ordinal α is trichotomous if the ∈-relation on α is trichotomous, that is to say (∀β ∈ α)(∀γ ∈ α)[β ∈ γ ∨ β = γ ∨ γ ∈ β]. Dana Scott posed the question whether it would be consistent with IZF to assume that the class of ordinals on which trichotomy

3 holds comprises a set rather than a proper class. A positive answer was given by D. McCarty and G. Rosolini (see [16], chapter 3, Lemma 9.5). Utilizing a realizability structure V(Kl) they showed that IZF plus the assertion ∃x ∀α[α ∈ x ↔ Tri(α)] gives rise to a theory equiconsistent with IZF. This equiconsistency result, however, appears to be peculiar to IZF. It will be shown that CZFR,E + 0 ∃x ∀α[α ∈ x ↔ Tri(α)] proves the consistency of CZF while CZFR + fREA + ∃x ∀α[α ∈ x ↔ Tri(α)] proves the consistency of CZF + REA.

0 2 Inductive definitions in CZFR In this section it will be shown that certain special inductive definitions 0 can be formalized on the basis of CZFR. Let CZFR result from CZF by replacing Strong Collection with Replacement. While CZF0 is particularly suited to accommodating inductively defined classes (see [2], section 4.2), CZFR does not seem to allow for formalizing such classes. To illustrate the obstacles posed to Replacement and overcome by Strong Collection, let Γ be a definable monotone operation on sets, that is: Γ(X) is set for all sets X, and whenever X,Y are sets such that X ⊆ Y then Γ(X) ⊆ Γ(Y ). We aim at defining a smallest class IΓ such that for all sets X ⊆ IΓ, Γ(X) ⊆ IΓ holds. The obvious way to do this is to iterate Γ through the ordinals, that is by defining [ Γα = Γ( Γβ) β∈α S α and letting IΓ = α Γ . Of course, IΓ can be formalized on the basis of CZFR. But what happens if we want to show that IΓ is closed under Γ? So suppose X ⊆ IΓ, where X is a set. Then for each u ∈ X there exists an ordinal α such that u ∈ Γα. With classical logic, we can always select the smallest ordinal such that u ∈ Γα, and then use Replacement (and other axioms) to find an ordinal β such that ∀u ∈ X ∃α ∈ β u ∈ Γα; whence S β β Γ(X) ⊆ Γ( α∈β Γ ) = Γ ⊆ IΓ, using the monotonicity of Γ. Strong Collection also enables one to find such an ordinal β. However, when the reasoning is based solely on intuitionistic logic, it is (in general) not possible to associate with u ∈ X a unique ordinal α such u ∈ Γα. Therefore it appears that Replacement fails to supply us with such an ordinal β and consequently we are left without any means of showing that IΓ is closed under Γ.

Deﬁnition 2.1 An ordinal is a transitive set whose elements are transitive also. As per usual, we use variables α, β, γ, . . . to range over ordinals. An ordinal α is said to be trichotomous or linear if ∀β ∈ α ∀γ ∈ α [β ∈ γ ∨ β = γ ∨ γ ∈ β].

4 We will use Tri(α) to abbreviate that α is trichotomous.

Deﬁnition 2.2 If R is a binary relation, that is a set of ordered pairs, we write xRy for hx, yi ∈ R. A partial ordering is a pair hA, Ri such that R partially orders A - that is, A is a set, R is a binary relation, R is transitive on A: ∀x, y, z ∈ A [xRy ∧ yRz → xRz], and R is irreﬂexive on A: ∀x ∈ A ¬xRx. Note that we do not assume R ⊆ A×A. So if hA, Ri is a partial ordering, then for any B ⊆ A, hB,Ri is a partial ordering. hA, Ri is a total ordering if it is a partial ordering and trichotomy holds: ∀x, y ∈ A [xRy ∨ x = y ∨ yRx],

Deﬁnition 2.3 If hA, Ri is a partial ordering and f : A → α is a function from A onto an ordinal α such that

(∀u ∈ A) f(u) = {f(v) | vRu ∧ v ∈ A}, then we will convey this by writing f : hA, Ri =∼ α. Note also that if hA, Ri is a total ordering and f : hA, Ri =∼ α, then α is trichotomous.

0 ∼ Lemma 2.4 (CZFR) Let hA, Ri be a partial ordering such that f : hA, Ri = α. Then the following induction principle holds for hA, Ri:

(∀u ∈ A) [(∀v ∈ A)(vRu → φ(v)) → φ(u)] → (∀u ∈ A) φ(u).

Proof: Assume

(∀u ∈ A) [(∀v ∈ A)(vRu → φ(v)) → φ(u)]. (3)

For β ∈ α deﬁne ψ(β) by

(∀z ∈ A)[f(z) = β → φ(z)].

Suppose that β ∈ α and (∀γ ∈ β) ψ(γ). Suppose β = f(x) for some x ∈ A. If u ∈ A and uRx, then f(u) ∈ β and hence φ(u). Thus (3) yields φ(x), and therefore ψ(β). Hence, by ∈-induction on α, we arrive at (∀β ∈ α) ψ(β); whence (∀u ∈ A) φ(u). 2

0 ∼ Corollary 2.5 (CZFR) Let hA, Ri be a partial ordering such that f : A = α and g : A =∼ β. Then α = β and f = g.

Proof: Using the induction principle of Lemma 2.4 one readily shows that f(u) = g(u) for all u ∈ A; whence α = β and f = g. 2

5 0 Theorem 2.6 (CZFR) If hA, Ri is a partial ordering then there is a smallest class X such that for all a ∈ A, whenever ∀u ∈ A (uRa → u ∈ X) then a ∈ X. This class will be denoted by WF(A, R) (the well-founded part of hA, Ri).

Proof: Suppose hA, Ri is a partial ordering. For u ∈ A let Au := {v ∈ A | vRu}. Let ∼ I = {u ∈ A | ∃α ∃f (f : hAu,Ri = α)}. (4)

Suppose a ∈ A and (∀u ∈ A)(uRa → u ∈ I). We want to show a ∈ I. By Corollary 2.5 we get that for all u ∈ Aa there exist a unique ordinal αu and a ∼ unique function fu such that fu : hAu,Ri = αu. Now let α = {αu | u ∈ Aa} and define f : Aa → α by f(u) = αu. The existence of α and f is owed to Replacement. In order to show that α is an ordinal it suffices to show that α is transitive. So let β ∈ αu, where u ∈ Aa. Then β = fu(z) for some z ∈ Au. The transitivity of R implies that fu is defined on Az, and thus by Corollary 2.5 we have fu ¹ Az = fz. As a result, β = fu(z) = {fu(v) | v ∈ Az} = {fz(v) | v ∈ Az} = αz; whence β ∈ α. Similarly it follows from ∼ Corollary 2.5 that f : hAa,Ri = α, and hence a ∈ I. Finally we have to show that I is the smallest class that satisfies the above closure property. So assume that X is a class such that for all y ∈ A, whenever ∀u ∈ A (uRy → u ∈ X) then y ∈ X. If a ∈ I then ∼ f : hAa,Ri = α for some α and f. Moreover, if x ∈ A and xRa then ∼ x ∈ I and g : hAx,Ri = β for some β ∈ α, where g is the restriction of f to Ax. Thence, proceeding by induction on α, one easily verifies that a ∈ X. 2

Deﬁnition 2.7 Suppose hA, Ri is a partial ordering and φ(u) is a set- theoretic formula. Let

ProghA,Ri(φ) iﬀ (∀u ∈ A) [(∀v ∈ A)(vRu → φ(v)) → φ(u)].

If X is a class {u | φ(u)} we also write ProghA,Ri(X) instead of ProghA,Ri(φ).

Corollary 2.8 Let hA, Ri be a partial ordering and φ(u) be a set-theoretic formula.

(i) (∀x ∈ A)[x ∈ WF(A, R) ↔ (∀u ∈ A)(uRx → u ∈ WF(A, R))].

(ii) ProghA,Ri(φ) → (∀x ∈ WF(A, R)) φ(x).

Proof: (i) and (ii) are immediate consequences of Theorem 2.6. 2

6 3 Regular extension axioms

The ﬁrst large set axiom proposed in the context of constructive set theory was the Regular Extension Axiom, REA, which Aczel introduced to accom- modate inductive deﬁnitions in Constructive Zermelo-Fraenkel Set Theory, CZF (cf. [1], [3]).

Deﬁnition 3.1 For sets A, B let AB be the class of all functions with domain A and with range contained in B. Let mv(AB) be the class of all sets R ⊆ A × B satisfying ∀u ∈ A ∃v ∈ B hu, vi ∈ R. The expression mv(AB) should be read as the collection of multi-valued functions from the set A to the set B. On the basis of CZF, AB is a set for all sets A, B, while, in general, it cannot be shown that mv(AB) is a set.

Deﬁnition 3.2 A set C is said to be regular if it is transitive, inhabited (i.e. ∃u u ∈ C) and for any u ∈ C and R ∈ mv(uC) there exists a set v ∈ C such that

∀x ∈ u ∃y ∈ v hx, yi ∈ R ∧ ∀y ∈ v ∃x ∈ u hx, yi ∈ R.

We write Reg(C) to express that C is regular. REA is the principle

∀x ∃y (x ⊆ y ∧ Reg(y)).

Deﬁnition 3.3 There are interesting weakened notions of regularity. A transitive inhabited set C is weakly regular if for any u ∈ C and R ∈ mv(uC) there exists a set v ∈ C such that

∀x ∈ u ∃y ∈ v hx, yi ∈ R.

We write wReg(C) to express that C is weakly regular. The Weak Regular Extension Axiom (wREA) is as follows: Every set is a subset of a weakly regular set. A transitive inhabited set C is functionally regular if for any u ∈ C and function f : u → C, ran(f) ∈ C. We write fReg(C) to express that C is functionally regular. The Functional Regular Extension Axiom (fREA) is as follows: Every set is a subset of a functionally regular set. S S Deﬁnition 3.4 A class A is said to be -closed if for all x ∈ A, x ∈ A. A class A is said to be closed under Exponentiation (Exp-closed) if for all x, y ∈ A, xy ∈ A. One is naturally led to considerS stronger notions of regularity, for in- stance that the set should also be -closed and Exp-closed. A transitive

7 S inhabited set C is said to be strongly regular if C is regular, -closed and Exp-closed. The Strong Regular Extension Axiom (sREA) is as follows: Every set is a subset of a strongly regular set.

Lemma 3.5 (CZF) If A is regular then A is weakly regular and functionally regular.

Proof: Obvious. 2 With respect to proof-theoretic strength, the axioms wREA, REA, and sREA are of the same caliber when added to CZF.

Theorem 3.6 (i) CZF has the same proof-theoretic strength as classical Kripke-Platek set theory (with the Inﬁnity axiom).

(ii) CZF + wREA has a (much) greater proof-theoretic strength than CZF. The theories CZF+wREA, CZF+REA, and CZF+sREA have the same proof-theoretic strength as the subsystem of second order 1 arithmetic with ∆2-Comprehension and Bar Induction.

Proof: See [23], Theorem 4.7. 2 A problem that was left open in [23] and that was also mentioned in [24] is whether CZF + fREA is of the same strength as the theories of Theorem 3.6(ii).

0 Lemma 3.7 (CZFR) Let C be a functionally regular such that 2 ∈ C. (i) C is closed under Pairing, that is to say, ∀x, y ∈ C {x, y} ∈ C.

(ii) If A ∈ C and f : A → C is a function from A to C, then f ∈ C.

Proof: (i): Given x, y ∈ C deﬁne a function g : 2 → C by g(0) = x and g(1) = y. Then {x, y} = ran(g) ∈ C. (ii): Assume A ∈ C and f : A → C. Then hx, f(x)i ∈ C for all x ∈ A by using (i) (twice). Thus x 7→ hx, f(x)i is a mapping from A to C whose range is f. By the functional regularity of C we thus get f ∈ C. 2

0 Corollary 3.8 CZFR + fREA proves Exponentiation. Proof: Given sets A, B, let C be a functionally regular set such that 2, A, B ∈ C. If f : A → B then f : A → C so that by Lemma 3.7, f ∈ C. As a result, AB = {f ∈ C | f : A → B}, and hence AB is a set by Bounded Separation. 2

8 0 Theorem 3.9 (CZFR+fREA) If hA, Ri is a partial ordering, then WF(A, R) is a set.

Proof: Suppose hA, Ri is a partial ordering. For u ∈ A let Au := {v ∈ A | vRu}. Note that by Replacement there is a function h with domain A such that h(u) = Au for all u ∈ A. Hence, owing to fREA, there exists a functionally regular set C such that 2, A, R ∈ C and Au ∈ C for all u ∈ A. Let ∼ IC = {u ∈ A | (∃α ∈ C)(∃f ∈ C)(f : hAu,Ri = α)}. (5)

IC is a set by Bounded Separation. Suppose a ∈ A and (∀u ∈ A)(uRa → u ∈ IC ). We want to show a ∈ IC . By Corollary 2.5 we get that for all u ∈ Aa there exist a unique ordinal αu ∈ C and a unique function fu ∈ C ∼ such that fu : hAu,Ri = αu. Now let α = {αu | u ∈ Aa} and deﬁne f : Aa → α by f(u) = αu. Since u 7→ αu is a mapping from Aa to C it follows from the functional regularity of C that α ∈ C. Thus, by 3.7, it ∼ follows that f ∈ C. Corollary 2.5 also guarantees that f : hAa,Ri = α, and hence a ∈ IC . In view of the proof of Theorem 2.6, the foregoing implies that IC = WF(A, R), whence WF(A, R) is a set. 2

0 4 A well-ordering proof in CZFR It is known that CZF has the same proof-theoretic strength as the classical theory of Kripke-Platek set theory, KP, and the theory of non-iterated inductive deﬁnitions ID1 (see, e.g., [24], Theorem 2.1(i)). The proof-theoretic ordinal of these theories is the so-called Bachmann-Howard ordinal. As Replacement is provable in CZF from Strong Collection, the Bachmann- 0 Howard ordinal is an upper bound for the proof-theoretic ordinal of CZFR. Let (B, ≺) be one of the familiar ordinal representation systems for the Bachmann-Howard ordinal, where B is a primitive recursive set of ordinal representations and ≺ denotes the primitive recursive ordering relation on B. Moreover, let (≺n)n∈N be a sequence of canonical initial segments of ≺ S 0 such that ≺= n∈N ≺n. That CZFR has the same proof-theoretic strength and the same stock of provably recursive functions as these theories can be 0 established, then, by showing that for ever n ∈ N, CZFR proves that ≺n is well-founded. In the following we will be assuming familiarity with the ordinal representation system for the Bachmann-Howard ordinal presented in [18]. We are to eliminate any uses of excluded middle and have to ensure 0 that the proofs can be carried out on the basis of CZFR. In the following, B and

9 of the set B(εΩ+1) and the corresponding ordering < on the representations, respectively, as codiﬁed in [18] §23. We shall reserve the variables a, b, c, . . . , x, y, z, u, v, . . . to range over elements of B.

Deﬁnition 4.1 (cf. [18], 23.30) An element a of B is said to be an additive principal number (notated by “a ∈ H”) if it can be represented in one of the forms a = ϕbc or a = ψb for some b, c ∈ B.

a =NF a1 + ··· + an iﬀ a = a1 + ··· + an ∧ a1, . . . , an ∈ H

∧ 0

a =NF ϕbc iﬀ a = ϕbc ∧ b, c

ψa ↓ iﬀ (∀x ∈ Ka)(x

Deﬁnition 4.2 Acc := WF(B ∩ Ω,

Lemma 4.3 (Acc-induction) Let θ(a) be a formula of set theory. Then we have the following induction principle:

(∀a ∈ Acc)[(∀b

Proof: This is an immediate consequence of Corollary 2.8(ii). 2

Lemma 4.4 (i) a ∈ Acc ∧ b

(ii) If a

Lemma 4.5 If a, b ∈ Acc then a + b ∈ Acc.

Proof: Let a ∈ Acc. We shall prove that a + b ∈ Acc for all b ∈ Acc using Acc-induction on b. So suppose that (∀c

Lemma 4.6 ∀x, y ∈ Acc (ϕxy ∈ Acc).

10 Proof: Let θ(a) be the formula ∀b ∈ Acc (ϕab ∈ Acc). We shall use induction on Acc. So assume a ∈ Acc and

(∀z

It suﬃces to show θ(a). Let b ∈ Acc and suppose

(∀y

We want to show ϕab ∈ Acc. The latter follows from

(8) is veriﬁed by induction on Gd.

Case 1 (d = ψd0): Then d ≤hb a or d ≤hb b. In either case d ∈ Acc since a, b ∈ Acc.

Case 2 (d =NF ϕd0d1): If d0

Case 3 (d =NF d1 + ··· + dn): Then the inductive assumption yields d1, . . . , dn ∈ Acc, so that d ∈ Acc by Lemma 4.5. The three foregoing cases furnish a proof of (8), and hence ϕab ∈ Acc. The upshot of the above is that we proved

(∀b ∈ Acc)[(∀y

Deﬁnition 4.7 For a ∈ B we deﬁne SCΩ(a) ⊆ B as follows:

1. SCΩ(0) = SCΩ(Ω) = ∅. S 2. SCΩ(a) = 1≤i≤n SCΩ(ai) if a =NF a1 + ··· + an.

3. SCΩ(a) = SCΩ(b) ∪ SCΩ(c) if a =NF ϕbc.

4. SCΩ(a) = {a} if a = ψa0. Let M := {a ∈ B | SC (a) ⊆ Acc} and deﬁne a

Prog (U):⇔ (∀y ∈ M)[(∀z

11 Lemma 4.8 Acc = M ∩ Ω.

Proof: Note that for every a ∈ B and x ∈ SCΩ(a) we have x

First suppose that a ∈ Acc. Since x ≤hb a holds for all x ∈ SCΩ(a), we have SCΩ(a) ⊆ Acc; hence a ∈ M ∩ Ω. Now let a ∈ M ∩ Ω. Then SCΩ(a) ⊆ Acc. By induction on Ga we verify a ∈ Acc. Obviously 0 ∈ Acc. If a =NF a1 + ··· + an, then a ∈ Acc follows from the inductive assumption and lemma 4.5. If a =NF ϕbc, then b, c ∈ Acc by the inductive assumption, and whence a ∈ Acc by Lemma 4.6. If a = ψb, then SCΩ(a) = {a} ⊆ Acc, so that a ∈ Acc. 2

Lemma 4.9 Let

Xψ := {a ∈ M | ψa ↓→ ψa ∈ Acc}.

Then we have ProgM(Xψ).

Proof: Note that ψa ↓ stands for (∀x ∈ Ka)(x

Deﬁnition 4.10 For X a class let

Xj := {b ∈ M | ∀a ∈ M [M ∩ a ⊆ X → M ∩ a + ωb ⊆ X]}. (9)

j Lemma 4.11 If X is a subclass of M and ProgM(X), then ProgM(X ).

j Proof: Assume (A) ProgM(X), (B) b ∈ M, and (C) ∀z

∀a ∈ M [M ∩ a ⊆ X → M ∩ a + ωb ⊆ X]. (10)

So suppose (D) a ∈ M ∧ M ∩ a ⊆ X, and let d ∈ M ∩ a + ωb.

If d

d = a + ωd1 + ··· ωdk .

12 As M ∩ a ⊆ X, we obtain M ∩ a + ωd1 ⊆ X from (C). Again, using (C), the latter implies M ∩ a + ωd1 + ωd2 ⊆ X, so that by repeating this kind of argument we arrive at M ∩ a + ωd1 + ··· + ωdk ⊆ X. In consequence, M ∩ d ⊆ X, so that by (A) and (D) we get d ∈ X. Since we have considered all the (decidable) cases, we have shown (10). 2

ηm Proposition 4.12 Let η1 := Ω + 1 and ηm+1 := ω . For every (meta) m and deﬁnable class X,

0 CZFR ` ProgM(X) → M ∩ ηm ⊆ X ∧ ηm ∈ X. (11)

Proof: We use (meta) induction on m. Assume ProgM(X). Then

∀a ∈ Acc [(∀u

0 j j j CZFR ` ProgM(X ) → M ∩ ηk ⊆ X ∧ ηk ∈ X , so that by 4.11,

0 j j CZFR ` ProgM(X) → M ∩ ηk ⊆ X ∧ ηk ∈ X . (12)

j Assume ProgM(X). Then also ProgM(X ) and from (12) we get M∩ηk +1 ⊆ Xj. The latter means that

b (∀b ∈ M ∩ ηk + 1)(∀a ∈ M)[M ∩ a ⊆ X → M ∩ a + ω ⊆ X].

Substituting 0 for a and ηk for b, the latter yields M ∩ ηm ⊆ X as ηm ∈ M. j A ﬁnal application of ProgM(X) gives M ∩ ηm + 1 ⊆ X . 2

0 Theorem 4.13 For every m, CZFR ` ηm ∈ Acc. The proof-theoretic 0 ordinal of CZFR is the Bachmann-Howard ordinal.

Proof: Let Xψ := {a ∈ M | ψa ↓ → ψa ∈ Acc}. Due to 4.9 we have 0 0 CZFR ` ProgM(Xψ) and hence, by 4.12, CZFR ` ψηm ∈ Acc. 2

Remark 4.14 The meta-induction of the previous result cannot be turned 0 into a formal induction within CZFR. The reason is that Xψ is a proper 0 class from the point of view of CZFR and therefore the sequence of classes

13 n 0 n+1 n j 0 Xψ with Xψ := Xψ and Xψ := (Xψ) is not deﬁnable in CZFR. On the n other hand, under the assumption that Acc is a set, one can show that Xψ is a set for all n ∈ ω, and hence

0 CZFR + ‘Acc is a set’ ` (∀n ∈ ω) ηn ∈ Acc. 0 And consequently, CZFR + ‘Acc is a set’ proves the consistency of CZF.

0 5 A well-ordering proof in CZFR + fREA

1 In their ordinal analysis of ∆2– CA + BI in [14], J¨agerand Pohlers used an ordinal representation system (OT,

14 1. a, b, c, d, e, f, x, y, z denote elements of OT. u, v, w denote elements of CT0. + 2. X ∩ u := {y ∈ X | Sy ≤ot u}. 3. φ ⊆ ψ iﬀ ∀x ∈ OT [φ(x) → ψ(x)]. 4. φ ∩ a ⊆ ψ iﬀ ∀x ∈ OT [x

(D1) ∀a ∈ X (a

+ X (D3) ∀u ∈ X (X ∩ u = Wu ). We use Ds(X) to express that X is a distinguished set; variables P,Q will always range over distinguished sets. Lemma 5.3 Let Q be a distinguished set. Let ψ(a) be a formula and let Prog(Q, ψ) stand for Prog(φ, ψ) with φ(a) being a ∈ Q. We then have:

(i) WF(Q,

15 Definition 5.4 Let Q be a distinguished set. We define a function FQ with domain Q by transfinite recursion over Q by   FQ(Sa) if a∈ / CT0 F (a) = WF({x ∈ OT | Sx = 0}, < ) if a = 0 Q  ot WF(M˜ a,

M˜ u = {x ∈ OT | Sx ≤ot u ∧ ∀v ∈ Wu [Kv ⊆ Wu]}. Note that the case distinctions involved in the foregoing definitions are decidable. Observe also that the definition draws on Theorem 3.9. Furthermore, in order for the latter to be a legitimate transfinite recursion over Q we must guarantee that, for u ∈ Q, if x ∈ Q and x

Q Q Lemma 5.5 Let Q be a distinguished set. If u ∈ Mu and ∀z ∈ Mu [z

Proof: This is a result from [8]. It is also proved in [19], Satz 10.8. The proof is intuitionistically sound as well. The case distinctions in these proofs concern the relation

Lemma 5.6 Let P and Q be distinguished sets. If u ∈ P ∪ Q and there exist a ∈ P and b ∈ Q such that u ≤ot a and u ≤ot b, then P ∩ u+ = Q ∩ u+. Proof: This, again, is a result from [8]. It is also proved in [19], Satz 10.10. The proof is intuitionistically sound. 2

∗ ∗ Lemma 5.7 If u ∈ L0, u ∈ FQ(u ), and Ds(FQ(u∗)), then u ∈ FQ(u) and Ds(FQ(u)).

∗ Proof: Suppose c

16 Lemma 5.8 Let u ∈ Q, u ∈ CT , u∈ / L0, and suppose

∀x ∈ Q [x

Then we have Ds(FQ(u)) and u ∈ FQ(u). S Proof: It follows from Lemma 5.6 that Wu = {FQ(u[x]) | x ∈ Q ∧ x

+ + Q ∩ t(u) = Wu ∩ t(u) . (13)

Now suppose c ∈ M˜ u and c

∀x

Since ∀v ∈ Q ∩ CT0 [v

u ∈ M˜ u. (15)

Using Lemma 5.5 we conclude from (14) and (15) that FQ(u) is a distinguished set and u ∈ FQ(u). 2

Proposition 5.9 If Q is a distinguished set and a ∈ Q, then FQ(a) is a distinguished set with a ∈ FQ(a). Proof: We will proceed by Q-induction on a (justiﬁed by Lemma 5.3(ii)). 1. If a = 0 then the assertion is obvious.

2. Suppose a∈ / CT0. Then Sa

17 Deﬁnition 5.10 Let [ W := {Q : Q is distinguished}.

For distinguished sets Q, P and x ∈ Q ∩ P , it follows from [13], Lemma 2.13 that (Q)x = (P )x. Thus we may deﬁne a set-valued operation C on W by letting

C(x) = FQ(x) for some Q with x ∈ Q. Corollary 5.11 For all a ∈ W, C(a) is a distinguished set.

Proof: This is an immediate consequence of Proposition 5.9. 2

Proposition 5.12 For every set-theoretic formula ψ(u), Prog(W, ψ) → ∀x ∈ W ψ(x). Proof: Suppose Prog(W, ψ) and a ∈ W. Then there exists a distinguished set Q such that a ∈ Q. Owing to Lemma 5.6, Prog(W, ψ) implies Prog(Q, ψ), so that by Lemma 5.3(ii) we obtain ψ(a). 2

Deﬁnition 5.13 If X is a subclass of W and a ∈ OT, let X ∩ a := {u ∈ X | x

Lemma 5.14 If M(a) and b ∈ W, then

1 1 ∀v ∈ W ∩ θ¯ ab [Kvθ¯ ab ⊆ W]. Proof: Lemma 5.6 and [13] Lemma 2.1(a) entail that for b ∈ W,

1 ∀v ∈ W ∩ θ¯ ab [Kvb ⊆ W]. By induction on Ga one gets that M(a) implies

1 ∀v ∈ W ∩ θ¯ ab [Kva ⊆ W].

1 1 As Kvθ¯ ab = Kva ∪ Kvb holds for v

18 Lemma 5.15 If B(a, b) then θ¯1ab ∈ W.

Proof: Suppose B(a, b). Let u = θ¯1ab. If u ∈ L0 the assumption yields u∗ ∈ W, so that by Lemma 5.7, θ¯1ab ∈ W. Now suppose u∈ / L0. B(a, b) in conjunction with [13] Lemma 1.2(c) and Lemma 2.2(a), then implies t(u) ∈ W, u∗ ∈ W, and ∀x ∈ W ∩ t(u)(u[x] ∈ W). B(a, b) also yields ∀v ∈ W ∩ u (Kvu ⊆ W) by Lemma 5.14. Hence, in view of Lemma 5.8, we get u ∈ W. 2

Proposition 5.16 Prog(M, R).

Proof: From Lemma 5.15 we conclude that

A(a) → ∀x[x ∈ W ∧ ∀u ∈ W ∩ x (θ¯1au ∈ W) → θ¯1ax ∈ W]. (16)

By Proposition 5.12, the latter entails

A(a) → ∀x ∈ W (θ¯1ax ∈ W), conﬁrming Prog (M, R). 2

Deﬁnition 5.17 For X a class let

X˜ j := {b | M(b) ∧ ∀a [M(a) ∧ M ∩ a ⊆ X → M ∩ a + ωb ⊆ X](17)}.

Lemma 5.18 If X is a subclass of {a | M(a)} and Prog(M,X), then Prog(M, X˜ j).

Proof: This is just like the proof of Lemma 4.11. 2

ωn(a) Proposition 5.19 Let ω0(a) := a and ωn+1(a) := ω . For each (meta) n and arbitrary set-theoretic formula φ(x) we have

0 CZFR + fREA ` Prog (M, φ) → φ(ωn(I + 1)). (18)

Proof: This follows from Lemma 5.18 by using (meta) induction on n similarly as in the proof of Proposition 4.12. 2

Theorem 5.20 For every (meta) n,

0 ¯0 ¯1 CZFR + fREA ` θ (θ ωn(I + 1)0)0 ∈ WF(OT,

19 0 Proof: By Propositions 5.16 and 5.19 we have CZFR + fREA ` R(ωn(I + 1)), so that 0 ¯1 CZFR + fREA ` θ ωn(I + 1)0 ∈ W as 0 ∈ W. Applying [13] Theorem 2.1(b), we arrive at

0 ¯0 ¯1 CZFR + fREA ` θ (θ ωn(I + 1)0)0 ∈ W. (20)

Since the elements of W less than θ¯100 form an initial segment, (20) entails (19). 2 Theorem 5.20 implies Theorem 5.1.

Remark 5.21 The meta-induction of the previous result cannot be turned 0 into a formal induction within CZFR+fREA or even CZF+REA. The reason is that {a | R(a)} is a proper class from the point of view of CZF+REA j and therefore the sequence of classes Rn with R0 := R and Rn+1 := R˜n is not deﬁnable in CZF + REA. On the other hand, under the assumption that W is a set, we have that {a | M(a)} is a set and thus the sequence (Rn)n∈ω becomes a sequence of sets. As a result,

0 ¯0 ¯1 CZFR + fREA + ‘W is a set’ ` (∀n ∈ ω) θ (θ ωn(I + 1)0)0 ∈ W.

0 And consequently, CZFR + fREA + ‘W is a set’ proves the consistency of CZF + REA.

6 Trichotomous ordinals

Dana Scott posed the question whether it is consistent with IZF to assume that the class of ordinals on which trichotomy holds comprises a set rather than a proper class. A positive answer was given by D. McCarty and G. Rosolini (see [16], chapter 3, Lemma 9.5). Utilizing a realizability structure V(Kl) they showed that IZF plus the assertion

∃x ∀α[α ∈ x ↔ Tri(α)] gives rise to a theory equiconsistent with IZF (actually, their proof yields that these theories are ≡ 0 in the sense of Deﬁnition 1.3). This equicon- Π2 sistency result, however, appears to be peculiar to IZF. In this section it will be shown that CZFR,E + ∃x ∀α[α ∈ x ↔ Tri(α)] proves the consis- 0 tency of CZF and that CZFR + fREA + ∃x ∀α[α ∈ x ↔ Tri(α)] proves the consistency of CZF + REA.

Proposition 6.1 CZFR,E + ∃x ∀α[α ∈ x ↔ Tri(α)] proves that Acc is a set.

20 Proof: Let S be the set of trichotomous ordinals and let hA, Ri be a total ordering. Recall from the proof of Theorem 2.6 that ∼ WF(A, R) = {u ∈ A | ∃α ∃f (f : hAu,Ri = α)}, where for u ∈ A, Au := {v ∈ A | vRu}. Since the relation R is trichotomous ∼ on A, f : hAu,Ri = α implies that α is trichotomous, and hence α ∈ S. Let C be the class [ [ {f | f : Au → α}. α∈S u∈A Mainly owing to Exponentiation, Replacement, and Union, C is a set in our background theory. Moreover, by the above, ∼ WF(A, R) = {u ∈ A | ∃α ∈ S ∃f ∈ C (f : hAu,Ri = α)}, so that WF(A, R) turns out to be a set. In particular, then, Acc is a set.2

0 Proposition 6.2 CZFR + fREA + ∃x ∀α[α ∈ x ↔ Tri(α)] proves that W is a set.

Proof: Let S be the set of trichotomous ordinals and let C be a functionally regular set with S ∈ C and

X X X + ∼ Wu = {x ∈ Mu | ∃α ∈ D ∃f ∈ D [f : hMu ∩ u ,

As a result of the above, X ⊆ OT is a distinguished set if and only if X ∈ C, ∀a ∈ X (a

+ X X + ∼ X ∩ u = {x ∈ Mu | ∃α ∈ D ∃f ∈ D [f : hMu ∩ u ,

Thence, by Bounded Separation, the class of distinguished sets is a set. 2

Corollary 6.3 (i) CZFR,E + ∃x ∀α[α ∈ x ↔ Tri(α)] proves the consistency of CZF.

0 (ii) CZFR + fREA + ∃x ∀α[α ∈ x ↔ Tri(α)] proves the consistency of CZF + REA.

21 Proof: (i) is a consequence of Proposition 6.1 and Remark 4.14 while (ii) is a consequence of Proposition 6.2 and Remark 5.21. 2 Naturally, one would like to know by how much the statement ∃x ∀α[α ∈ x ↔ Tri(α)] boosts the strength of CZF. The proof, though, that it does not boost the strength of IZF makes central use of the Powerset axiom and unbounded Separation. Unfortunately, the author of the present paper does not know of any upper bounds that do not involve the Powerset axiom.

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