Realizability for Constructive Zermelo-Fraenkel Set Theory

Annals of Mathematical Logic, 1982

We define recursive models of Martin-L6f's (type or) set theories. These models are a sort of recurs~ve realizabitlty; in fact, we show that for implieation-fl'ee formulae of HA% satisfaction in the model coincides with mr-HEO real~ability. Using an idea of AczeL we extend the model to a recursive model of the constructive set theories of Myhill and Friedman. Oar models can be described without presupposing any knowledge of Martin-L6f's theories, and may make them seem less mysterious. We use our models to ,btain several metamathematical resuRs, for example consistency and independence results concerning continuity of functions on compact metric spaces. On the other hand, Martin-L6f's (latest) theories refute continuity of functions from N ~ to N, a~ well as Church's thesis, although we show that all provably v'el!-defined functions are continuous. T(e, z, ]), and let k be larger than/'; and let y be a total index such that {y}(x) = 0 for x ~ n + 1, and {y}(n + 1)= k. Then {z}(y)= 1, while {z}(Ax0)= 0. Yet {y} and {Ax0} agree for their first m values, contradicting the supposition that {e}(z) was a modulus of continuity for e at AxO. That completes the proof of the iemma. [7] 1,3, The extended effective operations One natural idea is to extend the effective operations into transfinite types. Suppose, for example, that for each integer n, {e}(n)e E,t. Then e itself is in some sense an effective operation of type (He N)E~, regarding this as an infinite Cartesian product. To make this sort of construction systematically possible, we must assign codes to the types. Thus instead of 'm ~ E,' we will write where /~,, is a specific integer code which wil~ be assigned to E~. M here means the model of extended effective operations which we are constructing. We shall not give any definition o~' M by itself, but only of compound expressions containing M, such as M~:m~k~ We choose some (more or less arbitrary) functions as follows:

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik

Journal of Mathematical Sciences, 2006

LAS5. ϕ ⇒ (ψ ⇒ (ϕ ∧ ψ)); LAS6. ϕ ⇒ (ϕ ∨ ψ); LAS7. ψ ⇒ (ϕ ∨ ψ); LAS8. (ϕ ⇒ χ) ⇒ ((ψ ⇒ χ) ⇒ ((ϕ ∨ ψ) ⇒ χ)); LAS9. (ϕ ⇒ ψ) ⇒ ((ϕ ⇒ ¬ψ) ⇒ ¬ϕ); LAS10. (¬(¬ϕ)) ⇒ ϕ; LAS11. (∀vϕ) ⇒ ϕ(v θ) if v is a variable and θ is a term such that the substitution v θ in θ is admissible. LAS12. ϕ(v θ) ⇒ (∃v ϕ) under the same conditions as in LAS11; LAS13. (∀v(ψ ⇒ ϕ(v))) ⇒ (ψ ⇒ (∀vϕ)) if ψ does not contain a free variable v; LAS14. (∀v(ϕ(v) ⇒ ψ)) ⇒ ((∃v ϕ) ⇒ ψ) under the same condition as in LAS13. Proper axioms and axiom schemes cannot be formulated in the general case because they depend on a theory. The first-order theory which does not contain any proper axioms is called the first-order predicate calculus. Proper axioms and axiom schemes of LTS are stated in the next section. The deduction rules in the first-order theory are as follows: (1) the implication rule (≡ modus ponens (MP)): from ϕ and ϕ ⇒ ψ, it follows that ψ; (2) the generalization rule (Gen): from ϕ, it follows that ∀v ϕ. Let Φ be a totality of formulas, and let ψ be a formula of the theory T. A sequence f ≡ (ϕ i |i ∈ n + 1) ≡ (ϕ 0 ,. .. , ϕ n) of formulas of the theory T is called a deduction of the formula ψ from the totality Φ if ϕ n = ψ, and for any 0 < i ≤ n, one of following conditions holds: (1) ϕ i belongs to Φ; (2) there exist 0 ≤ k < j < i such that ϕ j is (ϕ k ⇒ ϕ i), i.e., ϕ i is obtained from ϕ k and ϕ k ⇒ ϕ i by the implication rule MP; (3) there exists 0 ≤ j < i such that ϕ i is ∀x ϕ j , where x is not a free variable of every formula from Φ, i.e., ϕ i is obtained from ϕ j by the generalization rule Gen with the given structural requirement. Denote this deduction either by f ≡ (ϕ 0 ,. .. , ϕ n) : Φ ψ, or by (ϕ 0 ,. .. , ϕ n) : Φ ψ, or by f : Φ ψ. A totality Φ a is called a totality of axioms of the theory T if Φ a consists of all explicit proper axioms of the theory T , all implicit proper axioms of the theory T , and all implicit logical axioms of the predicate calculus. If there exists a deduction f : Φ a ψ, then the formula ψ is said to be deducible in the axiomatic theory (T, Φ a), and the deduction f is said to be a proof of the formula ψ. A totality of formulas Φ is said to be contradictory (≡ non-consistent) if every formula of the theory T is deducible from it. In the opposite case, Φ is said to be noncontradictory (≡ consistent). An axiomatic theory (T, Φ a) is said to be contradictory (noncontradictory) if the totality of its axioms Φ a is contradictory (noncontradictory). Lemma 1. A totality of formulas Φ is contradictory if and only if the formulas ϕ and ¬ϕ for some sentence ϕ are deducible from Φ. ∃X (X ∈ a ∧ ∅ ∈ X ∧ ∀x (x ∈ X ⇒ (x ∪ a {x} a ∈ X))). Denote the postulated a-set by π. This axiom implies that the class ∅ is an a-set. By Axiom A5, ∅ is an α-set for every universe α. Consider the a-class κ ≡ {Y ∈ a | Y ⊂ π ∧ ∅ ∈ Y ∧ ∀y (y ∈ Y ⇒ (y ⊂ a ∧ y ∪ a {y} a ∈ Y))}. Since κ ⊂ P a (π), Axioms A9 and A8 imply that κ is an α-set. Consider the a-class ω ≡ {y ∈ a | ∀Y (Y ∈ κ ⇒ y ∈ Y)}. Since ω ⊂ π, by Axiom A8, we infer that ω is an a-set. Call it the a-set of natural numbers. By Axiom A5, ω is an α-set for every universe α. Consider the initial natural numbers 0 ≡ ∅, 1 ≡ 0 a {0} a , 2 ≡ 1 a {1} a ,. .. . From the definitions of sup{|V β | |β ∈ α} = |V α |. By the inequality which was proved above, we infer that |V α | ≤ κ. Suppose that |V α | = κ. Then by the regularity of the number κ, κ = ∪rng f implies κ ≤ α, but this contradicts the initial inequality α < κ. Therefore, |V α | < κ. Consequently, α ∈ C ⊂ C. By the principle of transfinite induction, C = On. Thus, C = κ. Lemma 15. If κ is an inaccessible cardinal, then κ = |V κ |.

Pacific Journal of Mathematics, 1980

1996

We show, that all Π 0 2-sentences, provable in the set theory KP I + U can be proved in Martin-Löf's Type Theory with W-type and one universe. Therefore set theoretical proofs can be considered as programs in type theory. The method used is a formalisation of proof theoretical methods in type theory. The result will be a high level type theory program using the full strength of Martin-Löf's Type Theory. We suggest the use of Kripke-Platek style set theory as a programming language.

2001

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form V κ where κ is a strongly inaccessible cardinal and V κ denotes the κ -th level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depends on the context in which they are embedded. The context here will be the theory CZF − of constructive Zermelo Fraenkel set theory but without ∈ -Induction (foundation). Let INAC be the statement that for every set there is an inaccessible set containing it. CZF − + INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF − + INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF − + INAC is a small ordinal known as the Feferman -Schütte ordinal Γ 0 .

Annali del Dipartimento di Filosofia, 2005

This is the first of two articles dedicated to the notion of con-structive set. In them we attempt a comparison between two different notions of set which occur in the context of the foun-dations for constructive mathematics. We also put them under perspective by stressing analogies ...

Archive for Mathematical Logic, 2007

First order Kleene realizability is given a semantic interpretation, including arithmetic and other types. These types extend at a stroke to full higher order intuitionistic logic. They are also useful themselves, e.g., as models for lambda calculi, for which see Asperti and Longo 1991 and papers on PERs and polymorphism (IEEE 1990). This semantics is simpler and more explicit than in Hyland 1982, giving the logical content of (1988) assemblies. Category theory here is only an organizing device and can be skipped, except in Lemmas 11-12 verifying the higher order logic. We verify some higher order constructive recursive analysis, and prove two metatheorems by generalizing the construction to other toposes.

Algebra and Logic, 1981