Normalization of IZF with Replacement

• Term Existence Property (TEP): If ∃x. φ(x) is provable, then φ(t) is provable for some term t. • Set Existence Property (SEP): If ∃x. φ(x) is provable, then there is a formula ψ(x) such that ∃!x. φ(x) ∧ ψ(x) is provable, where both φ and ψ are term-free. How to prove these properties for a given theory? There is a variety of methods applicable to constructive theories. Cut-elimination, proof normalization, realizability, Kripke models. . . . Normalization proofs, based on the Curry-Howard isomorphism principle, have the advantage of providing an explicit method of witness and program extraction from proofs. They also provide information about the behaviour of the proof system.

We are interested in intuitionistic set theory IZF. It is essentially what remains of ZF set theory after excluded middle is carefully taken away. An important decision to make on the way is whether to use Replacement or Collection axiom schema. We will call the version with Collection IZF C and the version with Replacement IZF R . In the literature, IZF usually denotes IZF C . Both theories extended with excluded middle are equivalent to ZF [Fri73].

They are not equivalent [FS85]. While the proof-theoretic power of IZF C is equivalent to that of ZF, the exact power of IZF R is unknown. Arguably IZF C is less constructive, as Collection, similarly to Choice, asserts the existence of a set without defining it.

Both versions have been investigated thoroughly. Results up to 1985 are presented in [Bee85,Ŝ85]. Later research was concentrated on weaker subsystems [AR01,Lub02]. A predicative constructive set theory CZF has attracted particular interest. [AR01] describes the set-theoretic apparatus available in CZF and provides further references.

We axiomatize IZF R , along with its intensional version IZF - R , using set terms. We define a typed lambda calculus λZ corresponding to proofs in IZF - R . We also define realizability for IZF - R , in the spirit of [McC84], and use it to show that λZ weakly normalizes. Strong normalization of λZ does not hold; moreover, we show that in non-well-founded IZF even weak normalization fails.

With normalization in hand, the properties NEP, DP and TEP easily follow. To show these properties for full, extensional IZF R , we define an inner model T of IZF R , consisting of what we call transitively L-stable sets. We show that a formula is true in IZF R iff its relativization to T is true in IZF - R . Therefore IZF R is interpretable in IZF - R . This allows us to use the properties proven for IZF - R . In IZF R , SEP easily follows from TEP. The importance of these properties in the context of computer science stems from the fact that they make it possible to extract programs from constructive proofs. For example, suppose IZF R ⊢ ∀n ∈ N∃m ∈ N. φ(n, m). From this proof a program can be extracted -take a natural number n, construct a proof IZF R ⊢ n ∈ N. Combine the proofs to get IZF R ⊢ ∃m ∈ N. φ(n, m) and apply NEP to get a number m such that IZF R ⊢ φ(n, m). A detailed account of program extraction from IZF R proofs can be found in [CM06].

There are many provers with the program extraction capability. However, they are usually based on variants of type theory, which is a foundational basis very different from set theory. This makes the process of formalizing program specification more difficult, as an unfamiliar new language and logic have to be learned from scratch. [LP99] strongly argues against using type theory for the specification purposes, instead promoting standard set theory.

IZF R provides therefore the best of both worlds. It is a set theory, with familiar language and axioms. At the same time, programs can be extracted from proofs. Our λZ calculus and the normalization theorem make the task of constructing the prover based on IZF R not very difficult.

This paper is mostly self-contained. We assume some familiarity with set theory, proof theory and programming languages terminology, found for example in [Kun80,SU06,Pie02]. The paper is organized as follows. We start by presenting in details intuitionistic first-order logic in section 2. In section 3 we define IZF R along with its intensional version IZF - R . In section 4 we define a lambda calculus λZ corresponding to IZF - R proofs. Realizability for IZF - R is defined in section 5. We use it to prove normalization of λZ in section 6, where we also show that non-well-founded IZF does not normalize. We prove the properties in section 7, and show how to derive them for full, extensional IZF R in section 8. Comparison with other results can be found in section 9.

Due to the syntactic character of our results, we present the intuitionistic first-o