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.
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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: rathjen@maths.leeds.ac.uk, Phone: +00 44 113 5109,
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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.
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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 Gödel’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, <) which is known to be decidable and incapable of defining the addition function.
2
A functional interpretation of ECST
In this section we will sketch a functional interpretation of ECST in a typed theory T−
∈ . A functional interpretation of CZF in T∈ - an extension of Gödel’s T to sets - was given by W. Burr. T−
∈ is a fragment of T∈
of [5] which arises from T∈ by firstly dropping the recursion terms Rσ and their defining axioms, and secondly
discarding the Foundation rule but adding the axioms of Strong Infinity as basic axioms. Since in a later section
T−
∈ will be interpreted in an admissible structure with urelements it is in order to recall the language and the
axioms of T−
∈.
Definition 2.1 (Definition of T−
∈ ) The collection T of linear type symbols is defined by: (1) o ∈ T , (2)
if σ, τ ∈ T then σ → τ ∈ T . The outermost brackets of a type symbol are usually suppressed. We use the
abbreviations 1 := o → o, 2 := 1 → o, σ → τ → ρ := σ → (τ → ρ) etc. T−
∈ contains the following basic
terms (by writing t : σ we convey that t is a term of type σ):
• countably many variables xσ , y σ , . . . : σ for each type σ
• constants 0, ω : o
• combinators Kτ σ : τ → σ → τ
• combinators Sρστ : (ρ → σ → τ ) → (ρ → σ) → ρ → τ
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M. Rathjen: Natural Numbers
• Suc , I : o → o → o
• N :o→o→o→o
• U : 1 → o → o.
The terms of T−
∈ are defined inductively as follows: Each basic term of type σ is a term of type σ; if t is a term
of type σ → τ and s is a term of type σ then (ts) is a term of type τ .
The ∆0 formulae of T−
∈ constitute the smallest collection of formulae that contains the atomic formulae s ∈ t,
s = t, ⊥ with s, t terms of type o and is closed under ∧, ∨, → and bounded quantification (∀x ∈ s), (∃x ∈ s),
where s, x : o and x does not occur in s. Note that ∆0 formulae do not contain equations of higher types but may
contain terms of arbitrary type as sub-terms.
The formulae of T−
∈ are generated from the ∆0 formulae and equations s = t between terms of the same type
σ (called equations of type σ) by closing off under ∧, → and bounded universal quantification (∀x ∈ s), where
s, x : o and x does not occur in s.
It appears to be opportune to point out that there are no unbounded quantifiers in T−
∈ formulae and that higher
type equations neither occur in the scope of a disjunction nor of an existential quantifier.
Definition 2.2 Below we shall assume that all terms have suitable types and that the formulae are well-formed.
We shall also drop the typing information in the combinators K and S. The Axioms and Rules of T−
∈ are the
following:
1. the intuitionistic rules for the propositional connectives and bounded quantifiers (for details see [5], §2);
2. equality axioms: s = s, s = t → ϕ(s) → ϕ(t), where s and t have the same type;
3. Set-Extensionality: (∀z ∈ s)z ∈ t ∧ (∀z ∈ t)z ∈ s → s = t
4. x ∈ 0 →⊥
5. x ∈ ω ↔ [x = 0 ∨ (∃y ∈ ω) x = Suc yy]
6. 0 ∈ s ∧ (∀x ∈ s) Suc xx ∈ s → (∀x ∈ ω)x ∈ s
7. x ∈ Suc st ↔ (x ∈ s ∨ x = t)
8. x ∈ Ist ↔ [x ∈ t ∧ (∀y ∈ s)x ∈ y]
9. x ∈ N str ↔ [x ∈ s ∧ (x ∈ t → x ∈ r)]
10. x ∈ U f s ↔ (∃y ∈ s)x ∈ f y
11. Kst = s
12. Sqtr = (qr)(tr)
13. (Extensionality rule): From θ → sa = ta infer θ → s = t,
providing a is an eigenvariable, i.e. a does not occur free in θ, s, t.
A more informal rendering of the axioms (7),
S (8), (9), (10) is the following: Suc st = s ∪ {t}, Ist = t ∩
N str = {x ∈ s | x ∈ t → x ∈ r}; U f s = {f x | x ∈ s}.
T
s,
×
The functional interpretation of T−
of [5],
∈ to be deployed is the same as in [5], namely the translation
−
Definition 4.1. That it works in the context of the weaker theories ECST and T∈ follows by carefully scouring
the proofs of [5].
Theorem 2.3 (Interpretation theorem) ECST plus Strong Collection is
×
-interpretable in T−
∈.
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P r o o f. Firstly it should be said that the constant ω of ECST is translated in the language of T−
∈ by replacing
−
×
it with the constant ω of T∈ . Note that the translation
does not affect ∆0 formulae and hence the axioms
of Strong Infinity are easily shown to be × -interpretable in T−
∈ by means of the axioms (4)-(7). For the other
axioms of ECST as well as Strong Collection the proof is the same as for [5], Theorem 4.3. Since ECST does
not have Set Induction the recursors Rσ are not required for the interpretation and thus it works with T−
∈ in lieu
of T∈ .
Corollary 2.4 Let ϕ(x, y) be a formula of ECST of the form ∃z θ(x, y, z) with θ ∆0 and all free variables
exhibited. Suppose that
ECST + Strong Collection ` ∃!y ϕ(x, y).
Then there are closed terms Q : 1, F : 1, Z : o → 1 of T−
∈ such that
T−
∈ ` (∃u ∈ Qx) u = u ∧ (∀u ∈ Qx) θ(x, F x, (Zu)x)
(1)
where x, u : o.
P r o o f. The same as for [5], Corollary 4.5.
Corollary 2.5 Let ψ(y) be a ∆0 formula of ECST with at most y free such that
ECST + Strong Collection ` ∃!y ψ(y).
Then there is a closed term p : o of T−
∈ such that
T−
∈ ` ψ(p).
P r o o f. Let θ(x, y, z) :≡ ψ(y) and make the substitution x 7→ 0 in (1). Finally let p := F 0.
3
ECST is a weak theory
The goal of this section is to prove the following theorem.
Theorem 3.1 ECST does not prove the existence of the addition function on ω. A fortiori ECST does not
prove small primitive recursion.
Let NL := (N; 0, SUC, <) be the structure obtained from the natural numbers by furnishing them with a
successor relation SUC such that SUC(n, m) ⇔ m = n + 1, a constant for the zero element and the less-than
relation. It is well known that the theory of NL is decidable and that the graph of the addition function on N is
not definable in NL (see [6], Section 3.2). Next we take a recursively saturated elementary extension M of NL ,
and finally we let HYPM be the smallest admissible set above the urelement structure M (as defined in [4], II.
Definition 5.8). HYPM is of the form
(M; L(M, λ) ∩ VM , ∈)
for some limit ordinal λ, where M stands for the domain of M, VM is the class of sets over M and L(M, λ) =
S
β<λ L(M, β) is a constructible hierarchy over M with L(M, 0) = M and L(M, α + 1) obtained from L(M, α)
by applying the Gödel functions F1 , . . . , F8 and some further simple functions to the elements of L(M, α) ∪
{L(M, α)}. The ordinal λ is usually denoted by O(M).
Since M is recursively saturated, it follows from a theorem of John Schlipf that O(M) = ω (see [4], IV.
Theorem 5.3). The urelement version of a theorem due to Gandy then implies that the relations on M in HYPM
are just the first-order definable relations of M (see [4], II. Corollary 7.2).
The next step is to use HYPM as a universe for interpreting ECST. There is an obstacle, though, as the set of
von Neumann integers ω is not an element of HYPM . To model the inductive set postulated by the strong infinity
axiom of ECST we will use the set M and view its ordering <M as an elementhood relation (notwithstanding
that it’s non-wellfounded). In order to satisfy extensionality we will introduce relations ∈˙ and =
˙ on the whole
of HYPM such that ∈˙ extends <M as well as ∈, and =
˙ extends = and satisfies
∀z ∈ aE ∃u ∈ bE z =
˙ u ∧ ∀u ∈ bE ∃z ∈ aE z =
˙ u ⇔ a=
˙ b
(2)
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M. Rathjen: Natural Numbers
for all a, b ∈ HYPM , where
aE =
a
{n ∈ M : n <M a}
if a is a set in HYPM ;
if a ∈ M .
We define the ∆1 predicate =
˙ by recursion (cf. [4], chapter I, Corollary 6.6) in the admissible set HYPM as
follows:
x=
˙ y
⇔
[x, y ∈ M ∧ x = y] ∨
[∀z ∈ xE ∃u ∈ yE (z =
˙ u) ∧ ∀u ∈ yE ∃z ∈ xE (z =
˙ u)].
Finally we let
x ∈˙ y
⇔
∃u ∈ yE (x =
˙ u).
Lemma 3.2 Let SUCM be the interpretation of SUC in M. Let a, b, c ∈ HYPM .
(i) a =
˙ a.
(ii) a =
˙ b ⇒ b=
˙ a.
(iii) a =
˙ b ∧ b=
˙ c ⇒ a=
˙ c.
(iv) x ∈ aE ⇒ x ∈˙ a.
(v) x ∈˙ a ∧ x =
˙ y ⇒ y ∈˙ a.
(vi) ∀z ∈ aE ∃u ∈ bE z =
˙ u ∧ ∀u ∈ bE ∃z ∈ aE z =
˙ u ⇔ a=
˙ b.
(vii) ∀z ∈˙ a ∃u ∈˙ b (z =
˙ u) ∧ ∀u ∈˙ b ∃z ∈˙ a (z =
˙ u) ⇔ a =
˙ b.
(viii) [x, y ∈ M ∧ x =
˙ y] ⇒ x = y.
(ix) [x, y ∈ M ∧ SUCM (x, y)] ⇒ xE ∪ {x} =
˙ y.
(x) 0 ∈˙ a ∧ ∀x, y ∈ M [x ∈˙ a ∧ SUCM (x, y) ⇒ y ∈˙ a] ⇒ ∀x ∈ M x ∈˙ a.
P r o o f. (i) is proved by induction on rk(a), the rank of a. (ii) is proved by induction on max(rk(a), rk(b)),
while (iii) is proved by induction on
max(rk(a), rk(b), rk(c)). (iv) follows from (i) and the definition of ∈˙ . (v) follows from (iii). (vi) follows from
(i)-(v). (vii) follows from (vi) and (iii). (viii): Note that {hx, yi ∈ M × M : x =
˙ y} is a set in HYPM and a
relation on M . Therefore it is definable in M and hence we can use induction on x along <M to show (vii). So
suppose that x =
˙ y. This entails that
∀u <M x ∃v <M y (u =
˙ v) ∧ ∀v <M y ∃u <M x (u =
˙ v).
By the inductive assumption, the latter implies
∀u <M x ∃v <M y (u = v) ∧ ∀v <M y ∃u <M x (u = v),
thus
∀u <M x (u <M y) ∧ ∀v <M y (v <M x),
whence x = y.
(ix) is immediate by definition.
(x). {x ∈ M | x ∈˙ a} is a set in HYPM and a subset of M , therefore it is definable in M. As a result, one can
use induction along <M to prove (x) (easily).
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3.1
7
Interpreting T−
∈ in HYPM
A function f with domain and range subsets of an admissible set A is said to be A-recursive if its graph is A-r.e.,
i.e. its graph is Σ1 on A.
We shall use the hereditarily =
˙ -extensional recursive functionals of finite type over HYPM to model T−
∈ in
HYPM . We shall need the following result.
Theorem 3.3 Let A = (B; A, ∈, . . .) be an admissible set. There is an A-r.e. relation Tn which parametrizes
the class of n-ary A-r.e. relations, with indices from A.
P r o o f. [4], V. Theorem 1.3.
Definition 3.4 In what follows it is assumed that all objects are in HYPM and that all quantifiers range over
HYPM . Let Tn be a HYPM -r.e. relation which parametrizes the n-ary HYPM -r.e. relations (as defined in the
proof of Theorem 3.3). If ∃!x Tn+1 (c, a1 , . . . , an , x) holds we shall write [c](a1 , . . . , an ) ↓ and also denote the
unique b such that Tn+1 (c, a1 , . . . , an , b) by [c](a1 , . . . , an ).
For each finite type σ we inductively define the =
˙ -extensional hereditarily recursive functionals of type σ
over HYPM as follows:
x ∈ To
x=
˙ oy
c ∈ Tρ→τ
:⇔ x ∈ HYPM
:⇔ x, y ∈ HYPM ∧ x =
˙ y
:⇔
∀a ∈ Tρ ([c](a) ↓ ∧ [c](a) ∈ Tτ ) ∧
∀aa0 ∈ Tρ [ a =
˙ ρ a0 → [c](a) =
˙ τ [c](a0 )]
c=
˙ ρ→τ d
:⇔ c, d ∈ Tρ→τ ∧ ∀a ∈ Tρ [c](a) =
˙ τ [d](a).
We remark that the classes Tσ and =
˙ σ are definable in HYPM . Of course, the complexity of the defining
formulas increases with the complexity of the type σ.
Definition 3.5 We define maps SucE , IE , NE , KE as follows:
SucE (a, b)
=
aE ∪ {b}
IE (a, b)
=
bE ∩ {x | ∀y ∈ aE x ∈ yE }
NE (a, b, c)
=
{x ∈ aE | x ∈ bE → x ∈ cE }
KE (a, b)
=
a.
Note that by Lemma 3.2 the foregoing maps are =
˙ -extensional in all arguments.
We also define relations UE , SE as follows:
[
UE (a, b, c) :⇔ c = {(f (x))E | x ∈ bE } for some function f with domain bE
such that ∀x ∈ bE T2 (a, x, f (x))
SE (a, b, c, d)
:⇔
∃xy [T2 (a, c, x) ∧ T2 (b, c, y) ∧ T2 (x, y, d)].
If there is a unique c such that UE (a, b, c) holds, this c will be denoted by UE (a, b). Similarly if there is exactly
one d such that SE (a, b, c, d) this d will be denoted by SE (a, b, c). Observe that UE is functional and total on
T1 × To and for a ∈ T1 we have
[
UE (a, b) = {([a](x))E | x ∈ bE }.
Similarly, SE is functional and total on Tρ→σ→τ × Tρ→σ × Tρ for all ρ, σ, τ , and if (a, b, c) ∈ Tρ→σ→τ ×
Tρ→σ × Tρ then SE (a, b, c) = ([a](c)) ([b](c)).
Observe also that UE is =
˙ 1 and =
˙ o extensional on T1 × To , i.e. if (a, b), (a0 , b0 ) ∈ T1 × To , a =
˙ 1 a0
0
0 0
0 0 0
and b =
˙ o b then UE (a, b) =
˙ o UE (a , b ). Likewise, SE is extensional in the sense that if (a, b, c), (a , b , c ) ∈
Tρ→σ→τ × Tρ→σ × Tρ , a =
˙ ρ→σ→τ a0 , b =
˙ ρ→σ b0 and c =
˙ ρ c0 then SE (a, b, c) =
˙ τ SE (a0 , b0 , c0 ).
On account of their definitions, all these maps and multi maps have HYPM -r.e. graphs and thus they have
indices in the sense of Theorem 3.3. Since an S-m-n or parameter theorem can easily be proved for A-r.e. relations for any admissible set A, one can construct indices eSuc , eI , eN , eU , eK , eS ∈ HYPM such that eSuc , eI ∈
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M. Rathjen: Natural Numbers
To→o→o , eN ∈ To→o→o→o , eU ∈ T1→o→o , eK ∈ Tτ →σ→τ for all τ, σ, and eS ∈ T(ρ→σ→τ )→(ρ→σ)→ρ→τ for
all ρ, σ, τ , and, moreover, [[eSuc ](a)](b) = SucE (a, b); [[eI ](a)](b) = IE (a, b); [[[eN ](a)](b)](c) = NE (a, b, c);
[[eU ](a)](b) = UE (a, b) if a ∈ T1 ; [[eK ](a)](b) = a; [[[eS ](a)](b)](c) = SE (a, b, c) whenever (a, b, c) ∈
Tρ→σ→τ × Tρ→σ × Tρ .
Theorem 3.6 T−
˙ -extensional hereditarily recursive functionals of type σ over
∈ has an interpretation in the =
HYPM as follows:
• The constants 0, ω, Kτ σ , Sρστ , Suc , I, N, U are interpreted as
0, M, eK , eS , eSuc , eI , eN , eU , respectively.
• Variables of type σ are interpreted as elements of Tσ .
• If s : σ → τ and t : σ are terms interpreted by elements s0 ∈ Tσ→τ and t0 ∈ Tσ , then st is interpreted as
[s0 ](t0 ).
• The elementhood relation is interpreted as ∈˙ while the equality between terms of type σ is interpreted as
=
˙ σ.
• Logical connectives and bounded quantifiers are interpreted by themselves.
Given a formula ϕ(xσ1 1 , . . . , xσnn ) of T−
∈ with all free variables exhibited, and a1 ∈ Tσ1 , . . . , an ∈ Tσn , we
denote by ϕ[a1 , . . . , an ]T the above interpretation with respect to the assignment xσi i 7→ ai . We then have:
σ1
T
σn
T−
∈ ` ϕ(x1 , . . . , xn ) ⇒ HYPM |= ϕ[a1 , . . . , an ] .
P r o o f. That the axioms of T−
∈ are validated by this interpretation follows from the choice of the constants
0, M, eK , eS , eSuc , eI , eN , eU , i.e. the defining equations of the corresponding maps KE , SE , SucE , IE , NE , UE
and the properties proved in Lemma 3.2. More precisely, axioms (5) and (7) follow from Lemma 3.2 (ix) while
axiom (6) follows from Lemma 3.2 (x).
Corollary 3.7 ECST plus Strong Collection does not prove the existence of the addition function on ω.
P r o o f. Assuming otherwise, ECST proves the statement ∃!f θ(f ), where θ(f ) is the formula expressing
that f is a function with domain ω × ω and range ω satisfying
∀x ∈ ω f (x, 0) = x ∧ ∀xy ∈ ω f (x, y + 1) = f (x, y) + 1
with y + 1 := y ∪ {y} and 0 := ∅. By Corollary 2.5 there then exists a closed term p : o of T−
∈ such that
T
`
θ(p),
and
hence
by
Theorem
3.6
we
conclude
that
HYP
|=
θ(p)
,
entailing
that
there
exists
a function
T−
M
∈
g ∈ HYPM with domain M × M and range M satisfying
∀n ∈ M g(n, 0) = n ∧ ∀nkk 0 ∈ M [SUCM (k, k 0 ) ⇒ SUCM (g(n, k), g(n, k 0 ))],
with SUCM being the interpretation of SUC in M. Since g ⊆ M × M and M is recursively saturated the graph
of g is definable in M. This implies that there exists a formula ψ(x, y, z, u1 , . . . , ur ) of the language of M and
m1 , . . . , mr ∈ M such that
M
|= ∀xy∃!z ψ(x, y, z, m)
~ ∧ ∀x ψ(x, 0, x, m)
~ ∧
(3)
∀xyzy 0 z 0 [ψ(x, y, z, m)
~ ∧ SUC(y, y 0 ) ∧ ψ(x, y 0 , z 0 , m)
~ ⇒ SUC(z, z 0 )] .
Abbreviating the formula of (3) by χ(m),
~ it holds M |= ∃u1 . . . ur χ(u1 , . . . , ur ) and therefore we conclude that
NL |= ∃u1 . . . ur χ(u1 , . . . , ur ) as M is an elementary extension of NL = (N; 0, SUC, <). But then addition
would be definable in NL , contradicting a well-known result about NL .
Corollary 3.8 s-ITERω is not provable in ECST plus Strong Collection.
P r o o f. Obvious by the previous Corollary.
Remark 3.9 Corollary 3.7 indicates that ECST plus Strong Collection is a very weak theory. We conjecture
that this theory has a finitistic consistency proof, say in elementary recursive arithmetic.
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4
9
CZF− is Π02 conservative over HA
In this section we show that CZF− is of the same proof-theoretic strength as Heyting Arithmetic, especially
that CZF− is Π02 conservative over HA. To be more precise, a Π02 statement θ of HA, i.e. a sentence of
the form ∀x∃yψ with ψ quantifier-free, has a canonical translation θs into the language of set theory, whereby
the quantifiers become restricted to ω and the symbols for the less-than relation, 0, successor, addition and
multiplication are replaced by their set-theoretic counterparts/descriptions. We will show that CZF− ` θs if and
only if HA ` θ (or PA ` θ). As in the previous section we shall use the method of recursively saturated models,
though this time a syntactic translation (hence finitistic reduction) is readily available, making the employment
of recursively saturated models an act of laziness.
The interpretations also validate some choice principles.
Definition 4.1 Let xRy stand for hx, yi ∈ R. A mathematically very useful axiom to have in set theory is the
Dependent Choices Axiom, DC, i.e., for all sets A and (set) relations R ⊆ a × a, whenever
(∀x ∈ A) (∃y ∈ A) xRy
and b0 ∈ A, then there exists a function f : ω → A such that f (0) = b0 and
(∀n ∈ ω) f (n)Rf (n + 1).
The Presentation Axiom, PAx, is an example of a choice principle which is validated upon interpretation in type
theory. In category theory it is also known as the existence of enough projective sets. A set P is a base if for
any P -indexed family (Xa )a∈P of inhabited sets Xa , there exists a function f with domain P such that, for all
a ∈ P , f (a) ∈ Xa . PAx is the statement that every set is the surjective image of a base.
Throughout this section we fix a countable recursively saturated model of PA M. In the language of arithmetic
we can define Turing machine application {e}(x) ' y i.e. the Turing machine with code e run on input number
x yields the result y. As M is a non-standard model there will be (codes of) non-standard Turing machines. For
e, x ∈ M we will use the shorthand e • x ↓ to convey that M |= {e}(x) ' y for some y ∈ M ; for a set X we use
e • x ∈ X to convey that M |= e • x ' y and y ∈ X for some y ∈ M (actually unique y).
We shall define “internal” versions of intensional and extensional transfinite type structures with dependent
products and dependent sums over M.
Definition 4.2 Let x, y 7→ (x, y) be an M-definable bijective pairing function on M with inverses z 7→ (z)0
and z 7→ (z)1 , i.e. ((x, y))0 = x and ((x, y))1 = y. Let (x, y, z) = (x, (y, z)) etc. 0, 1, 2, 3 will denote the first
four elements of M .
The intensional types of M and their elements are defined inductively. The set of elements of a type A is called
its extension and denoted by Â.
1. N
M
:= (0, 0) is a type with extension M .
M
2. For each m ∈ M , Nm := (0, SUCM (m)) is a type with extension {k ∈ M | k <M m}.
3. If A and B are types, then A +M B := (1, A, B) is a type with extension
{(0, x) | x ∈ Â} ∪ {(1, x) | x ∈ B̂}.
4. If A is a type and for each x ∈ Â, F (x) is a type, where F ∈ M and F (x) means F • x, then
M
Y
F (x) := (2, A, F )
x:A
[
is a type with extension {f ∈ M | ∀x ∈ Â f • x ∈ F
(x)}.
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10
M. Rathjen: Natural Numbers
5. If A is a type and for each x ∈ Â, F (x) is a type, where F ∈ M , then
M
X
F (x) := (3, A, F )
x:A
[
is a type with extension {(x, u) | x ∈ Â ∧ u ∈ F
(x)}.
The obvious question to ask is: Why should we distinguish between a type A and its extension Â. Well, the
reason is that we want to apply the Turing machine application operation of M to types. To make this possible,
types have to be elements of M .
Definition 4.3 We also define the extensional types of M. Here every type A comes equipped with its own
equality relation =A and functions between types have to respect those equality relations. Again, the set of
elements of a type A will be called its extension and be denoted by Â.
1. N
M
M
is a type with extension M . =N is just the equality of M.
M
2. For each m ∈ M , Nm is a type with extension {k ∈ M | k <M m}. =NM is just the equality of M
m
M
restricted to the extension of Nm .
3. If A and B are types, then A +M B is a type with extension
{(0, x) | x ∈ Â} ∪ {(1, x) | x ∈ B̂}.
The equality on A +M B is defined by
(i, x) =A+M B (j, y)
iff
[i = j = 0 ∧ x =A y] ∨ [i = j = 1 ∧ x =B y].
4. If A is a type, F ∈ M , and for each x ∈ Â, F (x) (= F • x) is a type such that F (x) and F (y) have the
same extension whenever x =A y, then then F is said to be a family of types over A.
5. If A is a type and F is a family of types over A, then
M
Y
F (x)
x:A
is a type with extension
[
{f ∈ M | ∀x ∈ Â f • x ∈ F
(x) ∧ ∀x, y ∈ Â[x =A y → f • x =F (x) f • y]}.
For f, g in the extension of
f =QM
x:A
g
QM
iff
F (x)
x:A
F (x),
∀x ∈ Â f • x =F (x) g • x.
6. If A is a type and F is a family of types over A, then
M
X
F (x)
x:A
[
is a type with extension {(x, u) | x ∈ Â ∧ u ∈ F
(x)}. Equality on
(u, v) =PM
x:A
F (x)
(w, z)
iff
PM
x:A
F (x) is defined by
u =A w ∧ v =F (u) z.
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11
Remark 4.4 The ordinary product and arrow types can be defined with the aid of dependent products and
sums, respectively. Let A, B be types and F ∈ M be a function such that F (x) = B for all x ∈ M .
A × B :=
M
M
X
Y
A → B :=
F (x)
x:A
F (x).
x:A
.
M
Definition 4.5 (The set-theoretic universe Vi ) Starting from the intensional type structure of M, we are
going to construct a universe of sets for intuitionistic set theory. The rough idea is that a set X is given by a type
A together with a set-valued function f defined on A (or rather the extension of A) such that X = {f (x) | x ∈ A}.
Again, the objects of this universe will be coded as elements of M . The above set will be coded as sup(A, f ),
where sup(A, f ) = (8, (A, f )) or whatever. We sometimes write {f (x) | x ∈ A} for sup(A, f ).
By the recursion theorem we can pick a standard number u such that {u}(x) ' sup(x, u) (this is provable in
PA).
M
The universe of sets over the intensional type structure of M, Vi , is defined inductively by two rules:
M
M
• sup(Nm , u) ∈ Vi for all m ∈ M ;
M
M
• if A is a type of M, f ∈ M , and ∀x ∈ Â f • x ∈ Vi , then sup(A, f ) ∈ Vi .
M
M
We shall use variables α, β, γ, . . . to range over elements of Vi . Each α ∈ Vi
Define ᾱ := A and α̃ := f .
M
We assign an ordinal rank(α) to every α ∈ Vi by letting
M
rank(sup(Nm , u))
=
rank(α)
=
is of the form sup(A, f ).
0
[
( {rank(α̃ • x) | x ∈ ᾱ}) + 1
M
if α is not of the form sup(Nm , u).
M
Whence if α is not of the form sup(Nm , u) then rank(α) > 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 ᾱ we’ll just write x ∈ ᾱ instead of
ˆ . In what follows we shall also omit •, i.e. e • x gets shortened to ex. ex1 x2 stands for (ex1 )x2 , ex1 x2 x3
x ∈ ᾱ
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
iff ∀i ∈ ᾱ [e0,0 i ∈ β̄ ∧ e0,1 i M α̃i = β̃(e0,0 i)] ∧
∀i ∈ β̄ [e1,0 i ∈ ᾱ ∧ e1,1 i M β̃i = α̃(e1,0 i)]
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M. Rathjen: Natural Numbers
For other formulas realizability is defined as follows:
e M α ∈ β
iff (e)0 ∈ β̄ ∧ (e)1 M α = β̃(e)0
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 ∈ ᾱ ei M φ(α̃i)
e M ∃x ∈ α φ(x) iff (e)0 ∈ ᾱ ∧ (e)1 M φ(α̃(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
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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 ∈ ᾱ ψ is an
abbreviation for ∀i, j ∈ ᾱ[i = j ∈ ᾱ → 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
∀i = j ∈ ᾱ [d0,0 i = e0,0 j ∈ β̄ ∧ d0,1 i = e0,1 j M α̃i = β̃(d0,0 i)] ∧
ξ
∀i = j ∈ β̄ [d1,0 i = e1,0 j ∈ ᾱ ∧ d1,1 i = e1,1 j M β̃i = α̃(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
iff (d)0 = (e)0 ∈ β̄ ∧ (d)1 = (e)1 M α = β̃(d)0
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 ∈ ᾱ → di = ej M φ(α̃i)]
ξ
ξ
d = e M ∃x ∈ α φ(x) iff (d)0 = (e)0 ∈ ᾱ ∧ (d)1 = (e)1 M φ(α̃(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 (ᾱ), 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
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15
proof for ε0 given in [9], §14. Let hA, ≺, 0, +̇, ξ 7→ ω̇ ξ i be a primitive recursive ordinal representation system
for ε0 with A ⊆ ω, ≺ being the ordering, and +̇ and ξ 7→ ω̇ ξ being the operations of addition and exponentiation
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
Sp(X) := {α | ∀ξ [∀η(η ≺ ξ → η ∈ X) → ∀η (η ≺ ξ +̇ω̇ α → η ∈ X)]}
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 ≺ ω̇ 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
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