The Natural Numbers in Constructive Set Theory - University of Leeds
Mathematical Logic Quarterly, 23 October 2009
The Natural Numbers in Constructive Set Theory Michael Rathjen1, ∗ 1
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
Received XXXX, revised XXXX, accepted XXXX Published online XXXX Key words Constructive set theory, Natural number object, recursively saturated models, functional interpretation, proof-theoretic strength MSC (2000) 03F50; 03F25; 03E55; 03B15; 03C70 Constructive set theory started with Myhill’s seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories. Copyright line will be provided by the publisher
1
Introduction
In a joint book project [3] (based on [2]), Peter Aczel and the author of this paper develop an extensive presentation of an approach to constructive mathematics that is based on an explicitly described axiom system. One of the aims of is to initiate an account of how constructive mathematics can be developed on the basis of a set theoretical axiom system. The intent is to prove each basic result relying on as weak an axiom system as possible. One of the first tasks to be addressed is the axiomatization of the natural numbers. The basic system with which [3] commences is called Elementary Constructive Set Theory, ECST. It is obtained from intuitionistic Zermelo-Fraenkel set theory, IZF by the following changes. 1. It uses the Replacement Scheme instead of the Collection Scheme. 2. It drops the Powerset Axiom and the Set Induction Scheme. 3. It uses the Bounded Separation Scheme instead of the full Separation Scheme. 4. It uses the Strong Infinity axiom instead of the Infinity axiom. Strong Infinity ∃a[Ind(a) ∧ ∀b[Ind(b) → ∀x ∈ a(x ∈ b)]] where we use the following abbreviations. • Empty(y) for (∀z ∈ y)⊥, • Succ(x, y) for ∀z[z ∈ y ↔ z ∈ x ∨ z = x], • Ind(a) for (∃y ∈ a)Empty(y) ∧ (∀x ∈ a)(∃y ∈ a)Succ(x, y). Some Consequences of ECST Among other things, in ECST one can show the existence of ordered pairs, Cartesian products, quotients and much more. Also, if ∀x ∈ a ∃!y φ(x, y) then there exists a unique function f with dom(f ) = a such that ∀x ∈ a φ(x, f (x)). The set of natural numbers will be obtained from the Strong Infinity axiom. The role of the number zero is played by the empty set. The infinite set of the Strong Infinity axiom is uniquely determined by its properties. ∗
Corresponding author
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M. Rathjen: Natural Numbers
Lemma 1.1 (ECST) Let θ(a) be the formula Ind(a) ∧ ∀y[Ind(y) → a ⊆ y]. If θ(a) and θ(b) then a = b. P r o o f. Ind(a) and Ind(b) yield a ⊆ b and b ⊆ a, hence a = b by Extensionality. Definition 1.2 The unique set a such that Ind(a) ∧ ∀y[Ind(y) → a ⊆ y] will be denoted by ω. We use a+ to denote a ∪ {a}. Theorem 1.3 (ECST) 1. ∀n ∈ ω [n = 0 ∨ (∃m ∈ ω) n = m+ ]. 2. ∀n ∈ ω (0 6= n+ ). 3. φ(0) ∧ ∀n ∈ ω[φ(n) → φ(n+ )] → (∀n ∈ ω) φ(n) for every bounded formula φ(n). 4. ∀n ∈ ω (n is transitive). 5. ∀n ∈ ω (n ∈ / n). 6. ∀n, m ∈ ω [n ∈ m → n+ ∈ m ∨ n+ = m]. 7. ∀n, m ∈ ω [n+ = m+ → n = m]. 8. ∀n ∈ ω (0 ∈ n+ ) 9. ∀n, m ∈ ω [n ∈ m ∨ n = m ∨ m ∈ n]. 10. m ∈ n ∨ m ∈ / n and m = n ∨ m 6= n for all n, m ∈ ω. P r o o f. [3] Theorem 6.3. The previous theorem entails that the structure (ω, 0, S) satisfies the Dedekind-Peano axioms, where S(n) = n+ = n ∪ {n} for n ∈ ω. Dedekind showed that from his axioms one could derive the following method for defining functions on ω (identifying N and ω) by iteration. Definition 1.4 (Small Iteration) For each set A, each F : A → A and each a0 ∈ A there is a unique function H : ω → A such that H(0) H(S(n))
= a0 , = F (H(n)).
We call this Small Iteration, abbreviated s-ITERω , because we require A to be a set. We get full Iteration by allowing A and F to be classes. By ∆0 -ITERω we will denote the schema where A and F are allowed to be ∆0 classes. In the next section it will be shown that ECST is a very weak theory in which Small Iteration cannot be proved. In particular it will be shown that the addition function on ω cannot be proved to exist in ECST. A familiar generalization of Iteration is Primitive Recursion. The set version is the following axiom. Definition 1.5 (Small Primitive recursion) For sets A, B, if F0 : B → A and F : B × ω × A → A then there is a (necessarily unique) H : B × ω → A such that for all b ∈ B H(b, 0) = F0 (b) H(b, n+ ) = F (b, n, H(b, n)) for all n ∈ ω We refer to this scheme as s-PRIMω . Note that s-ITERω is essentially a restricted version of s-PRIMω where B is a singleton set and F does not depend on its first argument. Copyright line will be provided by the publisher
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Theorem 1.6 (ECST) Assuming s-ITERω the axiom scheme s-PRIMω holds. P r o o f. [3] Theorem 6.17. Theorem 1.7 Heyting arithmetic, HA, can be interpreted in ECST + s-ITERω . P r o o f. Using s-PRIMω we see that the primitive recursive functions on ω can all be defined. Hence the fact that HA can be interpreted in ECST + s-ITERω follows from Theorem 1.3 and Theorem 1.6. Although s-ITERω gives us all the primitive recursive functions, in [3] s-ITERω has not been selected as the right axiom to complete the axiomatization of the natural numbers. This status has been bestowed on the next axiom. Definition 1.8 (Finite Powers Axiom, FPA) For each set A the class nA of functions from n to A is a set for all n ∈ ω. Note that this axiom is an immediate consequence of the Exponentiation Axiom and so is a theorem of CZF. Theorem 1.9 (ECST) The Finite Powers Axiom implies s-ITERω . P r o o f. [3] Theorem 6.10. There are several desirable consequences that FPA has but s-ITERω doesn’t seem to have (see [3]). Conjecture 1.10 ECST + s-ITERω does not prove FPA. With ECST + s-ITERω we have already reached the strength of Peano Arithmetic. In section 3 it will be shown that the addition of Strong Collection and Subset Collection to ECST doesn’t yield any more prooftheoretic strength. The latter system will be referred to as CZF− . In the main, it differs from CZF only by the omission of Set Induction. Moreover, adding the Axiom of Dependent Choices or the Presentation Axiom to CZF− doesn’t add proof-theoretic strength either. The final schema we are going to consider is ∆0 -ITERω . ∆0 -ITERω implies FPA on the basis of ECST (see [3]). The implication cannot be reversed though as the final section provides a proof that ECST + ∆0 -ITERω proves the consistency of PA. ∆0 -ITERω implies that every set possesses a transitive closure. It doesn’t seem to be possible to prove this from FPA. The proof of the weakness of ECST is established in two steps. Firstly, in section 2, ECST gets subjected to a functional interpretation in a version of G¨odel’s T over sets, dubbed T− ∈ . The second step, carried out in section 3, consists of interpreting T− in a type structure over a recursively saturated elementary extension of the ∈ structure (N; 0, SUC, 0. An essential characteristic of set theory is extensionality, i.e. that sets having the same elements are to be M identified. So if {f (x) | x ∈ A} and {g(y) | y ∈ B} are in Vi and for every x ∈ A there exists y ∈ B such that f (x) and g(y) represent the same set and conversely for every y ∈ B there exists x ∈ A such that f (x) and g(y) represent the same set, then {f (x) | x ∈ A} and {g(y) | y ∈ B} should be identified as sets. This idea gives rise M to an equivalence relation (bisimulation) on Vi . M
Definition 4.6 (Kleene realizability over Vi ) We will introduce a realizability semantics for sentences of set M theory with parameters from Vi . Bounded set quantifiers will be treated as quantifiers in their own right, i.e., M bounded and unbounded quantifiers are treated as syntactically different kinds of quantifiers. Let α, β ∈ Vi and e, f ∈ M . We write ei,j for ((e)i )j . To convey that x is in the extension of α ¯ we’ll just write x ∈ α ¯ instead of ˆ x∈α ¯ . In what follows we shall also omit •, i.e. e • x gets shortened to ex. ex1 x2 stands for (ex1 )x2 , ex1 x2 x3 stands for ((ex1 )x2 )x3 etc. For ordinals a, b we denote by a]b the natural ordinal sum (see e.g. [9], Definition 7.13). We define M
M
e M sup(Nm , u) = sup(Nm0 , u) iff m = m0 . If rank(α)] rank(β) > 0 let e M α = β
˜ 0,0 i)] ∧ iff ∀i ∈ α ¯ [e0,0 i ∈ β¯ ∧ e0,1 i M α ˜ i = β(e ˜ =α ∀i ∈ β¯ [e1,0 i ∈ α ¯ ∧ e1,1 i M βi ˜ (e1,0 i)] Copyright line will be provided by the publisher
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M. Rathjen: Natural Numbers
For other formulas realizability is defined as follows: e M α ∈ β
˜ 0 iff (e)0 ∈ β¯ ∧ (e)1 M α = β(e)
e M φ ∧ ψ
iff (e)0 M φ ∧ (e)1 M ψ iff (e)0 = 0 ∧ (e)1 M φ ∨ (e)0 = 1 ∧ (e)1 M ψ
e M φ ∨ ψ e M ¬φ e M φ → ψ
iff ∀f ∈ M ¬f M φ iff ∀f ∈ M f M φ → ef M ψ
e M ∀x ∈ α φ(x) iff ∀i ∈ α ¯ ei M φ(˜ αi) e M ∃x ∈ α φ(x) iff (e)0 ∈ α ¯ ∧ (e)1 M φ(˜ α(e)0 ) M
e M ∀xφ(x)
iff ∀α ∈ Vi eα M φ(α)
e M ∃xφ(x)
iff (e)0 ∈ Vi
M
∧ (e)1 M φ((e)0 ).
The definition of e M α = β falls under the scope of definition by transfinite recursion. Here it proceeds by recursion on rank(α)] rank(β). Theorem 4.7 ϕ(v1 , . . . , vr ) be a formula of set theory with at most the free variables exhibited. If CZF− + DC ` ϕ(v1 , . . . , vr ) M
then there exists e ∈ M such that for all α1 , . . . , αr ∈ Vi , M |= eα1 . . . αr ↓ and eα1 . . . αr M ϕ(α1 , . . . , αr ). e can be effectively constructed from the CZF− + DC-deduction of ϕ(v1 , . . . , vr ). P r o o f. Up to now we haven’t used the assumption that M is recursively saturated. Clearly the definition of Vi can be done in HYPM as it falls under the scope of Σ1 inductive definitions on an admissible set (see [4], VI. Theorem 3.8). One of the first axioms we have to find a realizer for is extensionality. If rank(α)] rank(β) > 0, and d M ∀x ∈ α x ∈ β ∧ ∀x ∈ β x ∈ α then clearly (by definition as it were) d M α = β, and thus M
i M ∀x ∈ α x ∈ β ∧ ∀x ∈ β x ∈ α → α = β,
(4)
where i is a machine code for the identity function. If, however, rank(α) = 0 and rank(β) = 0, we have to argue M M M differently. Then α = sup(Nm , u) and β = sup(Nk , u) for some m, k ∈ M . Put k ∗ := sup(Nk , u). One easily proves ∀d, k ∈ M [d M ∀x ∈ m∗ x ∈ k ∗ ∧ ∀x ∈ k ∗ x ∈ m∗
⇒ m = k]
(5)
for all m ∈ M by induction on <M . For this to be a legitimate induction though, the set of all m ∈ M such that (5) holds has to be definable in M. But as it is a set in HYPM and M is recursively saturated this is indeed the M case. The upshot of (4) and (5) is thus that (4) holds for all α, β ∈ Vi . M We also have to spell out which element of Vi is going to play the role of ω. Unsurprisingly, this will be M M ω := sup(N , j) with j an index for the function m 7→ sup(Nm , u). A consequence of (5) is that ω is injectively presented i.e. d M ω em = ω ek
⇒ m=k
(6)
holds for all d, k, m ∈ M . For the axiom of Strong Infinity one utilizes induction on <M for ∆1 formulas of HYPM which is legitimate on account of M’s recursive saturation. The proof of the theorem at issue proceeds by induction on the derivation of ϕ(v1 , . . . , vr ). Except for Extensionality and the role of ω (taken care of in the foregoing) the details are very similar to the proof of [10], Lemma 4.17. Induction on natural numbers therein has to be replaced by induction on <M and ∈-induction on sets has Copyright line will be provided by the publisher
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13 M
to be replaced by induction on rank(α) for α ∈ Vi . The validation of DC is similar to the validation of RDC in [10], Lemma 4.25, crucially exploiting (6). Corollary 4.8 Let θ be a Π02 sentence of arithmetic and θs be its set-theoretic rendering. If CZF− +DC ` θs then M |= θ. M
P r o o f. Put n∗ := sup(Nn , u). Let θ be the formula ∀x∃yϕ(x, y) with ϕ(x, y) quantifier-free. Then θs is the formula ∀x ∈ ω ∃y ∈ ω ϕ(x, y)s . From CZF− + DC ` θs we obtain e M ∀x ∈ ω ∃y ∈ ω ϕ(x, y)s for some e ∈ M . Unravelling the latter, we get ∀m ∈ M ∃e0 , k ∈ M e0 M ϕ(m∗ , k ∗ )s . The claim follows from the fact that e0 M ϕ(m∗ , k ∗ )s implies M |= ϕ(m, k). The details of proving this fact are too laborious and tedious and thus have to be omitted. Corollary 4.9 CZF− + DC is Π02 -conservative over PA and HA. P r o o f. By Corollary 4.8, if CZF− + DC ` θs for a Π02 statement θ, then M |= θ. Since M was an arbitrary recursively saturated model of PA and every countable model of PA has a recursively saturated elementary extension, θ holds in all countable models of PA and is thus provable in PA. Moreover, PA and HA prove the same Π02 statements. Corollary 4.10 The use of recursively saturated models is not necessary for establishing Corollary 4.9. Instead of using a translation of CZF− + DC into HYPM one can use a similar syntactic translation into the theory PArΩ of [7] which is conservative over PA, thus providing a finitistic reduction of CZF− + DC to PA and HA. Conjecture 4.11 We conjecture that CZF− is conservative over HA for all arithmetic formulae. M
M
We shall also consider an extensional version of Vi , dubbed Vξ , and extensional Kleene realizability over M
Vξ . M
Definition 4.12 (The set-theoretic universe Vξ ) Here we start from the extensional type structure of M. M
M
The universe of sets over the extensional type structure of M, Vξ , and an equality relation =VM on Vξ ξ
are defined inductively. Rather than x =A y we shall write x = y ∈ A. ∀x = y ∈ A ψ is an abbreviation for ∀x, y ∈ A[x =A y → ψ]. M The simultaneous inductive definition of Vξ and =VM has the following clauses: ξ
M
M
M
M
M
1. sup(Nm , u) ∈ Vξ and sup(Nm , u) = sup(Nm , u) ∈ Vξ for all m ∈ M . 2. Let A, B be extensional types of M and f, g ∈ M . M
M
M
(i) If ∀x ∈ A f x ∈ Vξ and ∀x = y ∈ A f x = f y ∈ Vξ , then sup(A, f ) ∈ Vξ . M
M
[(ii) If A and B have the same elements, sup(A, f ), sup(B, g) ∈ Vξ , and ∀x ∈ A f x = gx ∈ Vξ , then M
sup(A, f ) = sup(B, g) ∈ Vξ . M
Definition 4.13 (Extensional Kleene realizability over Vξ ) We write di,j for ((d)i )j . ∀i = j ∈ α ¯ ψ is an abbreviation for ∀i, j ∈ α ¯ [i = j ∈ α ¯ → ψ]. We define M
ξ
M
e = d M sup(Nm , u) = sup(Nm0 , u) iff m = m0 . If rank(α)] rank(β) > 0 let ξ
d = e M α = β
iff ξ
˜ 0,0 i)] ∧ ∀i = j ∈ α ¯ [d0,0 i = e0,0 j ∈ β¯ ∧ d0,1 i = e0,1 j M α ˜ i = β(d ξ ˜ =α ∀i = j ∈ β¯ [d1,0 i = e1,0 j ∈ α ¯ ∧ d1,1 i = e1,1 j M βi ˜ (d1,0 i)]
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M. Rathjen: Natural Numbers
For other formulas realizability is defined as follows: ξ
d = e M α ∈ β ξ
d = e M φ ∧ ψ ξ
d = e M φ ∨ ψ ξ
d = e M ¬φ
ξ ˜ 0 iff (d)0 = (e)0 ∈ β¯ ∧ (d)1 = (e)1 M α = β(d) ξ
iff
ξ
d = e M φ → ψ
ξ
iff (d)0 = (e)0 M φ ∧ (d)1 = (e)1 M ψ ξ iff (d)0 = (e)0 = 0 ∧ (d)1 = (e)1 M φ ξ ∨ (d)0 = (e)0 = 1 ∧ (d)1 = (e)1 M ψ
iff
ξ
∀f ∈ M ¬f = f M φ ξ ξ ∀f, g ∈ M f = g M φ → df = eg M ψ
ξ
ξ
d = e M ∀x ∈ α φ(x) iff ∀i, j [i = j ∈ α ¯ → di = ej M φ(˜ αi)] ξ
ξ
d = e M ∃x ∈ α φ(x) iff (d)0 = (e)0 ∈ α ¯ ∧ (d)1 = (e)1 M φ(α ˜ (d)0 ) M
ξ
d = e M ∀xφ(x)
M
ξ
d = e M ∃xφ(x) ξ
M
ξ
iff ∀α, β ∈ Vξ [α = β ∈ Vξ → dα = eβ M φ(α)] iff (d)0 = (e)0 ∈ Vξ
ξ
∧ (d)1 = (e)1 M φ((d)0 ).
ξ
e M θ iff e = e M θ. Theorem 4.14 Let ϕ(v1 , . . . , vr ) be a formula of set theory with at most the free variables exhibited. If CZF− + PAx ` ϕ(v1 , . . . , vr ) M
then there exists an e ∈ M such that for all α1 , . . . , αr ∈ Vξ , M |= eα1 . . . αr ↓ and
ξ
eα1 . . . αr M ϕ(α1 , . . . , αr ). e can be effectively constructed from the CZF− + PAx-deduction of ϕ(v1 , . . . , vr ). P r o o f. The CZF− part of the proof is the same as for Theorem 4.7. For the PAx part one first defines a M M M map τ : Extensional types of M → Vξ as in [1] Theorem 7.1 except that τ (N ) := sup(N , j) where j an M
M
M
index for the function m 7→ sup(Nm , u) and τ (Nm ) = sup(Nm , u). The function τ actually has an index eτ as M it can be defined by the recursion theorem in M. Next one shows that every τ (A) is realizably a base in Vξ and M
M
that every α ∈ Vξ is the image of the base S(τ (¯ α), α) as defined in [1] Theorem 7.3. Thus Vξ realizes PAx. More details can be found in [10], section 4.4. Corollary 4.15 CZF− + PAx is Π02 -conservative over PA and HA. Corollary 4.16 The use of recursively saturated models is not necessary for establishing Corollary 4.15. Instead of using a translation of CZF− + PAx into HYPM one can use a similar syntactic translation into the theory PArΩ of [7] which is conservative over PA, thus providing a finitistic reduction of CZF− + PAx to PA and HA.
5
ECST + ∆0 -ITERω is stronger than CZF−
Theorem 5.1 ECST + ∆0 -ITERω proves the consistency of CZF− . P r o o f. We know that CZF− is finitistically reducible to Heyting Arithmetic and Peano Arithmetic. Gentzen’s consistency proof of Peano Arithmetic uses an ordinal representation system for the ordinal ε0 and transfinite induction up to this ordinal for primitive recursive predicates. Apart from the transfinite induction, Gentzen’s proof is formalizable in primitive recursive arithmetic. It thus suffices to show that transfinite induction up to ε0 is provable in ECST + ∆0 -ITERω for arbitrary sets. For definiteness we shall now refer to the wellordering Copyright line will be provided by the publisher
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15
˙ ξ 7→ ω˙ ξ i be a primitive recursive ordinal representation system proof for ε0 given in [9], §14. Let hA, ≺, 0, +, ˙ and ξ 7→ ω˙ ξ being the operations of addition and exponentiation for ε0 with A ⊆ ω, ≺ being the ordering, and + with base ω. In what follows let X be a set. Variables α, ξ, η are assumed to range over A. The wellordering proof uses the Sprung (jump) operation ˙ ω˙ α → η ∈ X)]} Sp(X) := {α | ∀ξ [∀η(η ≺ ξ → η ∈ X) → ∀η (η ≺ ξ + and the ∆0 predicate Prog(≺, X) := ∀α [∀ξ(ξ ≺ α → ξ ∈ X) → α ∈ X.] Sp(X) is a set by Bounded Separation. Given a set X we can use ∆0 -ITERω to get a (unique) function FX with domain ω such that FX (0) = X and FX (n + 1) = Sp(FX (n)). By the same proof as for [9], Lemma 15.6 one proves that Prog(≺, X) → Prog(≺, Sp(X)).
(7)
Consequently with ∆0 induction on ω one gets Prog(≺, X) → ∀n ∈ ω Prog(≺, FX (n)).
(8)
By the same proof as for [9] Lemma 15.5 combined with (7) one obtains Prog(≺, X) ∧ ∀ξ[ξ ≺ α → ξ ∈ Sp(X)] → ∀ξ[ξ ≺ ω˙ α → ξ ∈ X].
(9)
(8) and (9) yield that Prog(≺, X) → ∀α α ∈ X i.e. transfinite induction up to ε0 for arbitrary sets. Acknowledgements This material is based upon work supported by the National Science Foundation under Award No. DMS-0301162. I am grateful to the referee for making a number of helpful suggestions.
References [1] P. Aczel: The type theoretic interpretation of constructive set theory: Choice principles. In: A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium (North Holland, Amsterdam 1982) 1–40. [2] P. Aczel, M. Rathjen: Notes on constructive set theory, Technical Report 40, Institut Mittag-Leffler (The Royal Swedish Academy of Sciences, 2001). http://www.mittag-leffler.se/preprints/0001/, Preprint No. 40. [3] P. Aczel, M. Rathjen: Notes on constructive set theory, Preprint (2006) 225 pages. (Available from the authors upon request.) [4] J. Barwise: Admissible Sets and Structures (Springer-Verlag, Berlin, Heidelberg, New York, 1975). [5] W. Burr: Functional Interpretation of Aczel’s constructive set theory. Annals of Pure and Applied Logic 104 (2000) 31–73. [6] H.B. Enderton: A Mathematical Introduction to Logic. Second Edition (Academic Press, London, 2001). [7] G. J¨ager: Fixed points in Peano arithmetic with ordinals. Annals of Pure and Applied Logic 60 (1993) 119–132. [8] J. Myhill: Constructive set theory. Journal of Symbolic Logic 40 (1975) 347–382. [9] W. Pohlers: Proof theory. Lecture Notes in Mathematics 1407 (Springer, Berlin, 1989). [10] M. Rathjen: The formulae-as-classes interpretation of constructive set theory. In: H. Schwichtenberg, K. Spies (eds.): Proof Technology and Computation (IOS Press, Amsterdam, 2006) 279–322.
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The Natural Numbers in Constructive Set Theory Michael Rathjen1, ∗ 1
Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT, England
Received XXXX, revised XXXX, accepted XXXX Published online XXXX Key words Constructive set theory, Natural number object, recursively saturated models, functional interpretation, proof-theoretic strength MSC (2000) 03F50; 03F25; 03E55; 03B15; 03C70 Constructive set theory started with Myhill’s seminal 1975 article [8]. This paper will be concerned with axiomatizations of the natural numbers in constructive set theory discerned in [3], clarifying the deductive relationships between these axiomatizations and the strength of various weak constructive set theories. Copyright line will be provided by the publisher
1
Introduction
In a joint book project [3] (based on [2]), Peter Aczel and the author of this paper develop an extensive presentation of an approach to constructive mathematics that is based on an explicitly described axiom system. One of the aims of is to initiate an account of how constructive mathematics can be developed on the basis of a set theoretical axiom system. The intent is to prove each basic result relying on as weak an axiom system as possible. One of the first tasks to be addressed is the axiomatization of the natural numbers. The basic system with which [3] commences is called Elementary Constructive Set Theory, ECST. It is obtained from intuitionistic Zermelo-Fraenkel set theory, IZF by the following changes. 1. It uses the Replacement Scheme instead of the Collection Scheme. 2. It drops the Powerset Axiom and the Set Induction Scheme. 3. It uses the Bounded Separation Scheme instead of the full Separation Scheme. 4. It uses the Strong Infinity axiom instead of the Infinity axiom. Strong Infinity ∃a[Ind(a) ∧ ∀b[Ind(b) → ∀x ∈ a(x ∈ b)]] where we use the following abbreviations. • Empty(y) for (∀z ∈ y)⊥, • Succ(x, y) for ∀z[z ∈ y ↔ z ∈ x ∨ z = x], • Ind(a) for (∃y ∈ a)Empty(y) ∧ (∀x ∈ a)(∃y ∈ a)Succ(x, y). Some Consequences of ECST Among other things, in ECST one can show the existence of ordered pairs, Cartesian products, quotients and much more. Also, if ∀x ∈ a ∃!y φ(x, y) then there exists a unique function f with dom(f ) = a such that ∀x ∈ a φ(x, f (x)). The set of natural numbers will be obtained from the Strong Infinity axiom. The role of the number zero is played by the empty set. The infinite set of the Strong Infinity axiom is uniquely determined by its properties. ∗
Corresponding author
E-mail: [email protected], Phone: +00 44 113 5109, Copyright line will be provided by the publisher
2
M. Rathjen: Natural Numbers
Lemma 1.1 (ECST) Let θ(a) be the formula Ind(a) ∧ ∀y[Ind(y) → a ⊆ y]. If θ(a) and θ(b) then a = b. P r o o f. Ind(a) and Ind(b) yield a ⊆ b and b ⊆ a, hence a = b by Extensionality. Definition 1.2 The unique set a such that Ind(a) ∧ ∀y[Ind(y) → a ⊆ y] will be denoted by ω. We use a+ to denote a ∪ {a}. Theorem 1.3 (ECST) 1. ∀n ∈ ω [n = 0 ∨ (∃m ∈ ω) n = m+ ]. 2. ∀n ∈ ω (0 6= n+ ). 3. φ(0) ∧ ∀n ∈ ω[φ(n) → φ(n+ )] → (∀n ∈ ω) φ(n) for every bounded formula φ(n). 4. ∀n ∈ ω (n is transitive). 5. ∀n ∈ ω (n ∈ / n). 6. ∀n, m ∈ ω [n ∈ m → n+ ∈ m ∨ n+ = m]. 7. ∀n, m ∈ ω [n+ = m+ → n = m]. 8. ∀n ∈ ω (0 ∈ n+ ) 9. ∀n, m ∈ ω [n ∈ m ∨ n = m ∨ m ∈ n]. 10. m ∈ n ∨ m ∈ / n and m = n ∨ m 6= n for all n, m ∈ ω. P r o o f. [3] Theorem 6.3. The previous theorem entails that the structure (ω, 0, S) satisfies the Dedekind-Peano axioms, where S(n) = n+ = n ∪ {n} for n ∈ ω. Dedekind showed that from his axioms one could derive the following method for defining functions on ω (identifying N and ω) by iteration. Definition 1.4 (Small Iteration) For each set A, each F : A → A and each a0 ∈ A there is a unique function H : ω → A such that H(0) H(S(n))
= a0 , = F (H(n)).
We call this Small Iteration, abbreviated s-ITERω , because we require A to be a set. We get full Iteration by allowing A and F to be classes. By ∆0 -ITERω we will denote the schema where A and F are allowed to be ∆0 classes. In the next section it will be shown that ECST is a very weak theory in which Small Iteration cannot be proved. In particular it will be shown that the addition function on ω cannot be proved to exist in ECST. A familiar generalization of Iteration is Primitive Recursion. The set version is the following axiom. Definition 1.5 (Small Primitive recursion) For sets A, B, if F0 : B → A and F : B × ω × A → A then there is a (necessarily unique) H : B × ω → A such that for all b ∈ B H(b, 0) = F0 (b) H(b, n+ ) = F (b, n, H(b, n)) for all n ∈ ω We refer to this scheme as s-PRIMω . Note that s-ITERω is essentially a restricted version of s-PRIMω where B is a singleton set and F does not depend on its first argument. Copyright line will be provided by the publisher
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Theorem 1.6 (ECST) Assuming s-ITERω the axiom scheme s-PRIMω holds. P r o o f. [3] Theorem 6.17. Theorem 1.7 Heyting arithmetic, HA, can be interpreted in ECST + s-ITERω . P r o o f. Using s-PRIMω we see that the primitive recursive functions on ω can all be defined. Hence the fact that HA can be interpreted in ECST + s-ITERω follows from Theorem 1.3 and Theorem 1.6. Although s-ITERω gives us all the primitive recursive functions, in [3] s-ITERω has not been selected as the right axiom to complete the axiomatization of the natural numbers. This status has been bestowed on the next axiom. Definition 1.8 (Finite Powers Axiom, FPA) For each set A the class nA of functions from n to A is a set for all n ∈ ω. Note that this axiom is an immediate consequence of the Exponentiation Axiom and so is a theorem of CZF. Theorem 1.9 (ECST) The Finite Powers Axiom implies s-ITERω . P r o o f. [3] Theorem 6.10. There are several desirable consequences that FPA has but s-ITERω doesn’t seem to have (see [3]). Conjecture 1.10 ECST + s-ITERω does not prove FPA. With ECST + s-ITERω we have already reached the strength of Peano Arithmetic. In section 3 it will be shown that the addition of Strong Collection and Subset Collection to ECST doesn’t yield any more prooftheoretic strength. The latter system will be referred to as CZF− . In the main, it differs from CZF only by the omission of Set Induction. Moreover, adding the Axiom of Dependent Choices or the Presentation Axiom to CZF− doesn’t add proof-theoretic strength either. The final schema we are going to consider is ∆0 -ITERω . ∆0 -ITERω implies FPA on the basis of ECST (see [3]). The implication cannot be reversed though as the final section provides a proof that ECST + ∆0 -ITERω proves the consistency of PA. ∆0 -ITERω implies that every set possesses a transitive closure. It doesn’t seem to be possible to prove this from FPA. The proof of the weakness of ECST is established in two steps. Firstly, in section 2, ECST gets subjected to a functional interpretation in a version of G¨odel’s T over sets, dubbed T− ∈ . The second step, carried out in section 3, consists of interpreting T− in a type structure over a recursively saturated elementary extension of the ∈ structure (N; 0, SUC, 0. An essential characteristic of set theory is extensionality, i.e. that sets having the same elements are to be M identified. So if {f (x) | x ∈ A} and {g(y) | y ∈ B} are in Vi and for every x ∈ A there exists y ∈ B such that f (x) and g(y) represent the same set and conversely for every y ∈ B there exists x ∈ A such that f (x) and g(y) represent the same set, then {f (x) | x ∈ A} and {g(y) | y ∈ B} should be identified as sets. This idea gives rise M to an equivalence relation (bisimulation) on Vi . M
Definition 4.6 (Kleene realizability over Vi ) We will introduce a realizability semantics for sentences of set M theory with parameters from Vi . Bounded set quantifiers will be treated as quantifiers in their own right, i.e., M bounded and unbounded quantifiers are treated as syntactically different kinds of quantifiers. Let α, β ∈ Vi and e, f ∈ M . We write ei,j for ((e)i )j . To convey that x is in the extension of α ¯ we’ll just write x ∈ α ¯ instead of ˆ x∈α ¯ . In what follows we shall also omit •, i.e. e • x gets shortened to ex. ex1 x2 stands for (ex1 )x2 , ex1 x2 x3 stands for ((ex1 )x2 )x3 etc. For ordinals a, b we denote by a]b the natural ordinal sum (see e.g. [9], Definition 7.13). We define M
M
e M sup(Nm , u) = sup(Nm0 , u) iff m = m0 . If rank(α)] rank(β) > 0 let e M α = β
˜ 0,0 i)] ∧ iff ∀i ∈ α ¯ [e0,0 i ∈ β¯ ∧ e0,1 i M α ˜ i = β(e ˜ =α ∀i ∈ β¯ [e1,0 i ∈ α ¯ ∧ e1,1 i M βi ˜ (e1,0 i)] Copyright line will be provided by the publisher
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M. Rathjen: Natural Numbers
For other formulas realizability is defined as follows: e M α ∈ β
˜ 0 iff (e)0 ∈ β¯ ∧ (e)1 M α = β(e)
e M φ ∧ ψ
iff (e)0 M φ ∧ (e)1 M ψ iff (e)0 = 0 ∧ (e)1 M φ ∨ (e)0 = 1 ∧ (e)1 M ψ
e M φ ∨ ψ e M ¬φ e M φ → ψ
iff ∀f ∈ M ¬f M φ iff ∀f ∈ M f M φ → ef M ψ
e M ∀x ∈ α φ(x) iff ∀i ∈ α ¯ ei M φ(˜ αi) e M ∃x ∈ α φ(x) iff (e)0 ∈ α ¯ ∧ (e)1 M φ(˜ α(e)0 ) M
e M ∀xφ(x)
iff ∀α ∈ Vi eα M φ(α)
e M ∃xφ(x)
iff (e)0 ∈ Vi
M
∧ (e)1 M φ((e)0 ).
The definition of e M α = β falls under the scope of definition by transfinite recursion. Here it proceeds by recursion on rank(α)] rank(β). Theorem 4.7 ϕ(v1 , . . . , vr ) be a formula of set theory with at most the free variables exhibited. If CZF− + DC ` ϕ(v1 , . . . , vr ) M
then there exists e ∈ M such that for all α1 , . . . , αr ∈ Vi , M |= eα1 . . . αr ↓ and eα1 . . . αr M ϕ(α1 , . . . , αr ). e can be effectively constructed from the CZF− + DC-deduction of ϕ(v1 , . . . , vr ). P r o o f. Up to now we haven’t used the assumption that M is recursively saturated. Clearly the definition of Vi can be done in HYPM as it falls under the scope of Σ1 inductive definitions on an admissible set (see [4], VI. Theorem 3.8). One of the first axioms we have to find a realizer for is extensionality. If rank(α)] rank(β) > 0, and d M ∀x ∈ α x ∈ β ∧ ∀x ∈ β x ∈ α then clearly (by definition as it were) d M α = β, and thus M
i M ∀x ∈ α x ∈ β ∧ ∀x ∈ β x ∈ α → α = β,
(4)
where i is a machine code for the identity function. If, however, rank(α) = 0 and rank(β) = 0, we have to argue M M M differently. Then α = sup(Nm , u) and β = sup(Nk , u) for some m, k ∈ M . Put k ∗ := sup(Nk , u). One easily proves ∀d, k ∈ M [d M ∀x ∈ m∗ x ∈ k ∗ ∧ ∀x ∈ k ∗ x ∈ m∗
⇒ m = k]
(5)
for all m ∈ M by induction on <M . For this to be a legitimate induction though, the set of all m ∈ M such that (5) holds has to be definable in M. But as it is a set in HYPM and M is recursively saturated this is indeed the M case. The upshot of (4) and (5) is thus that (4) holds for all α, β ∈ Vi . M We also have to spell out which element of Vi is going to play the role of ω. Unsurprisingly, this will be M M ω := sup(N , j) with j an index for the function m 7→ sup(Nm , u). A consequence of (5) is that ω is injectively presented i.e. d M ω em = ω ek
⇒ m=k
(6)
holds for all d, k, m ∈ M . For the axiom of Strong Infinity one utilizes induction on <M for ∆1 formulas of HYPM which is legitimate on account of M’s recursive saturation. The proof of the theorem at issue proceeds by induction on the derivation of ϕ(v1 , . . . , vr ). Except for Extensionality and the role of ω (taken care of in the foregoing) the details are very similar to the proof of [10], Lemma 4.17. Induction on natural numbers therein has to be replaced by induction on <M and ∈-induction on sets has Copyright line will be provided by the publisher
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13 M
to be replaced by induction on rank(α) for α ∈ Vi . The validation of DC is similar to the validation of RDC in [10], Lemma 4.25, crucially exploiting (6). Corollary 4.8 Let θ be a Π02 sentence of arithmetic and θs be its set-theoretic rendering. If CZF− +DC ` θs then M |= θ. M
P r o o f. Put n∗ := sup(Nn , u). Let θ be the formula ∀x∃yϕ(x, y) with ϕ(x, y) quantifier-free. Then θs is the formula ∀x ∈ ω ∃y ∈ ω ϕ(x, y)s . From CZF− + DC ` θs we obtain e M ∀x ∈ ω ∃y ∈ ω ϕ(x, y)s for some e ∈ M . Unravelling the latter, we get ∀m ∈ M ∃e0 , k ∈ M e0 M ϕ(m∗ , k ∗ )s . The claim follows from the fact that e0 M ϕ(m∗ , k ∗ )s implies M |= ϕ(m, k). The details of proving this fact are too laborious and tedious and thus have to be omitted. Corollary 4.9 CZF− + DC is Π02 -conservative over PA and HA. P r o o f. By Corollary 4.8, if CZF− + DC ` θs for a Π02 statement θ, then M |= θ. Since M was an arbitrary recursively saturated model of PA and every countable model of PA has a recursively saturated elementary extension, θ holds in all countable models of PA and is thus provable in PA. Moreover, PA and HA prove the same Π02 statements. Corollary 4.10 The use of recursively saturated models is not necessary for establishing Corollary 4.9. Instead of using a translation of CZF− + DC into HYPM one can use a similar syntactic translation into the theory PArΩ of [7] which is conservative over PA, thus providing a finitistic reduction of CZF− + DC to PA and HA. Conjecture 4.11 We conjecture that CZF− is conservative over HA for all arithmetic formulae. M
M
We shall also consider an extensional version of Vi , dubbed Vξ , and extensional Kleene realizability over M
Vξ . M
Definition 4.12 (The set-theoretic universe Vξ ) Here we start from the extensional type structure of M. M
M
The universe of sets over the extensional type structure of M, Vξ , and an equality relation =VM on Vξ ξ
are defined inductively. Rather than x =A y we shall write x = y ∈ A. ∀x = y ∈ A ψ is an abbreviation for ∀x, y ∈ A[x =A y → ψ]. M The simultaneous inductive definition of Vξ and =VM has the following clauses: ξ
M
M
M
M
M
1. sup(Nm , u) ∈ Vξ and sup(Nm , u) = sup(Nm , u) ∈ Vξ for all m ∈ M . 2. Let A, B be extensional types of M and f, g ∈ M . M
M
M
(i) If ∀x ∈ A f x ∈ Vξ and ∀x = y ∈ A f x = f y ∈ Vξ , then sup(A, f ) ∈ Vξ . M
M
[(ii) If A and B have the same elements, sup(A, f ), sup(B, g) ∈ Vξ , and ∀x ∈ A f x = gx ∈ Vξ , then M
sup(A, f ) = sup(B, g) ∈ Vξ . M
Definition 4.13 (Extensional Kleene realizability over Vξ ) We write di,j for ((d)i )j . ∀i = j ∈ α ¯ ψ is an abbreviation for ∀i, j ∈ α ¯ [i = j ∈ α ¯ → ψ]. We define M
ξ
M
e = d M sup(Nm , u) = sup(Nm0 , u) iff m = m0 . If rank(α)] rank(β) > 0 let ξ
d = e M α = β
iff ξ
˜ 0,0 i)] ∧ ∀i = j ∈ α ¯ [d0,0 i = e0,0 j ∈ β¯ ∧ d0,1 i = e0,1 j M α ˜ i = β(d ξ ˜ =α ∀i = j ∈ β¯ [d1,0 i = e1,0 j ∈ α ¯ ∧ d1,1 i = e1,1 j M βi ˜ (d1,0 i)]
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M. Rathjen: Natural Numbers
For other formulas realizability is defined as follows: ξ
d = e M α ∈ β ξ
d = e M φ ∧ ψ ξ
d = e M φ ∨ ψ ξ
d = e M ¬φ
ξ ˜ 0 iff (d)0 = (e)0 ∈ β¯ ∧ (d)1 = (e)1 M α = β(d) ξ
iff
ξ
d = e M φ → ψ
ξ
iff (d)0 = (e)0 M φ ∧ (d)1 = (e)1 M ψ ξ iff (d)0 = (e)0 = 0 ∧ (d)1 = (e)1 M φ ξ ∨ (d)0 = (e)0 = 1 ∧ (d)1 = (e)1 M ψ
iff
ξ
∀f ∈ M ¬f = f M φ ξ ξ ∀f, g ∈ M f = g M φ → df = eg M ψ
ξ
ξ
d = e M ∀x ∈ α φ(x) iff ∀i, j [i = j ∈ α ¯ → di = ej M φ(˜ αi)] ξ
ξ
d = e M ∃x ∈ α φ(x) iff (d)0 = (e)0 ∈ α ¯ ∧ (d)1 = (e)1 M φ(α ˜ (d)0 ) M
ξ
d = e M ∀xφ(x)
M
ξ
d = e M ∃xφ(x) ξ
M
ξ
iff ∀α, β ∈ Vξ [α = β ∈ Vξ → dα = eβ M φ(α)] iff (d)0 = (e)0 ∈ Vξ
ξ
∧ (d)1 = (e)1 M φ((d)0 ).
ξ
e M θ iff e = e M θ. Theorem 4.14 Let ϕ(v1 , . . . , vr ) be a formula of set theory with at most the free variables exhibited. If CZF− + PAx ` ϕ(v1 , . . . , vr ) M
then there exists an e ∈ M such that for all α1 , . . . , αr ∈ Vξ , M |= eα1 . . . αr ↓ and
ξ
eα1 . . . αr M ϕ(α1 , . . . , αr ). e can be effectively constructed from the CZF− + PAx-deduction of ϕ(v1 , . . . , vr ). P r o o f. The CZF− part of the proof is the same as for Theorem 4.7. For the PAx part one first defines a M M M map τ : Extensional types of M → Vξ as in [1] Theorem 7.1 except that τ (N ) := sup(N , j) where j an M
M
M
index for the function m 7→ sup(Nm , u) and τ (Nm ) = sup(Nm , u). The function τ actually has an index eτ as M it can be defined by the recursion theorem in M. Next one shows that every τ (A) is realizably a base in Vξ and M
M
that every α ∈ Vξ is the image of the base S(τ (¯ α), α) as defined in [1] Theorem 7.3. Thus Vξ realizes PAx. More details can be found in [10], section 4.4. Corollary 4.15 CZF− + PAx is Π02 -conservative over PA and HA. Corollary 4.16 The use of recursively saturated models is not necessary for establishing Corollary 4.15. Instead of using a translation of CZF− + PAx into HYPM one can use a similar syntactic translation into the theory PArΩ of [7] which is conservative over PA, thus providing a finitistic reduction of CZF− + PAx to PA and HA.
5
ECST + ∆0 -ITERω is stronger than CZF−
Theorem 5.1 ECST + ∆0 -ITERω proves the consistency of CZF− . P r o o f. We know that CZF− is finitistically reducible to Heyting Arithmetic and Peano Arithmetic. Gentzen’s consistency proof of Peano Arithmetic uses an ordinal representation system for the ordinal ε0 and transfinite induction up to this ordinal for primitive recursive predicates. Apart from the transfinite induction, Gentzen’s proof is formalizable in primitive recursive arithmetic. It thus suffices to show that transfinite induction up to ε0 is provable in ECST + ∆0 -ITERω for arbitrary sets. For definiteness we shall now refer to the wellordering Copyright line will be provided by the publisher
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˙ ξ 7→ ω˙ ξ i be a primitive recursive ordinal representation system proof for ε0 given in [9], §14. Let hA, ≺, 0, +, ˙ and ξ 7→ ω˙ ξ being the operations of addition and exponentiation for ε0 with A ⊆ ω, ≺ being the ordering, and + with base ω. In what follows let X be a set. Variables α, ξ, η are assumed to range over A. The wellordering proof uses the Sprung (jump) operation ˙ ω˙ α → η ∈ X)]} Sp(X) := {α | ∀ξ [∀η(η ≺ ξ → η ∈ X) → ∀η (η ≺ ξ + and the ∆0 predicate Prog(≺, X) := ∀α [∀ξ(ξ ≺ α → ξ ∈ X) → α ∈ X.] Sp(X) is a set by Bounded Separation. Given a set X we can use ∆0 -ITERω to get a (unique) function FX with domain ω such that FX (0) = X and FX (n + 1) = Sp(FX (n)). By the same proof as for [9], Lemma 15.6 one proves that Prog(≺, X) → Prog(≺, Sp(X)).
(7)
Consequently with ∆0 induction on ω one gets Prog(≺, X) → ∀n ∈ ω Prog(≺, FX (n)).
(8)
By the same proof as for [9] Lemma 15.5 combined with (7) one obtains Prog(≺, X) ∧ ∀ξ[ξ ≺ α → ξ ∈ Sp(X)] → ∀ξ[ξ ≺ ω˙ α → ξ ∈ X].
(9)
(8) and (9) yield that Prog(≺, X) → ∀α α ∈ X i.e. transfinite induction up to ε0 for arbitrary sets. Acknowledgements This material is based upon work supported by the National Science Foundation under Award No. DMS-0301162. I am grateful to the referee for making a number of helpful suggestions.
References [1] P. Aczel: The type theoretic interpretation of constructive set theory: Choice principles. In: A.S. Troelstra and D. van Dalen, editors, The L.E.J. Brouwer Centenary Symposium (North Holland, Amsterdam 1982) 1–40. [2] P. Aczel, M. Rathjen: Notes on constructive set theory, Technical Report 40, Institut Mittag-Leffler (The Royal Swedish Academy of Sciences, 2001). http://www.mittag-leffler.se/preprints/0001/, Preprint No. 40. [3] P. Aczel, M. Rathjen: Notes on constructive set theory, Preprint (2006) 225 pages. (Available from the authors upon request.) [4] J. Barwise: Admissible Sets and Structures (Springer-Verlag, Berlin, Heidelberg, New York, 1975). [5] W. Burr: Functional Interpretation of Aczel’s constructive set theory. Annals of Pure and Applied Logic 104 (2000) 31–73. [6] H.B. Enderton: A Mathematical Introduction to Logic. Second Edition (Academic Press, London, 2001). [7] G. J¨ager: Fixed points in Peano arithmetic with ordinals. Annals of Pure and Applied Logic 60 (1993) 119–132. [8] J. Myhill: Constructive set theory. Journal of Symbolic Logic 40 (1975) 347–382. [9] W. Pohlers: Proof theory. Lecture Notes in Mathematics 1407 (Springer, Berlin, 1989). [10] M. Rathjen: The formulae-as-classes interpretation of constructive set theory. In: H. Schwichtenberg, K. Spies (eds.): Proof Technology and Computation (IOS Press, Amsterdam, 2006) 279–322.
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