Typed lambda-calculus in classical Zermelo-Frænkel set theory Jean-Louis Krivine U.F.R. de Math´ematiques, Universit´e Paris VII 2 Place Jussieu 75251 Paris cedex 05 e-mail [email protected] In this paper, we develop a system of typed lambda-calculus for the Zermelo-Frænkel set theory, in the framework of classical logic. The first, and the simplest system of typed lambda-calculus is the system of simple types, which uses the intuitionistic propositional calculus, with the only connective !. It is very important, because the well known Curry- Howard correspondence between proofs and programs was originally discovered with it, and because it enjoys the normalization property : every typed term is strongly normal- izable. It was extended to second order intuitionistic logic, in 1970, by J.-Y. Girard[4], under the name of system F, still with the normalization property. More recently, in 1990, the Curry-Howard correspondence was extended to classical logic, following Felleisen and Griffin[6] who discovered that the law of Peirce corresponds to control instructions in functional programming languages. It is interesting to notice that, as early as 1972, Clint and Hoare[1] had made an analogous remark for the law of excluded middle and controlled jump instructions in imperative languages. There are now many type systems which are based on classical logic ; among the best known are the system LC of J.-Y. Girard[5] and the λµ-calculus of M. Parigot[11]. We shall use below a system closely related to the latter, called the λc-calculus[8, 9]. Both systems use classical second order logic and have the normalization property. In the sequel, we shall extend the λc-calculus to the Zermelo-Frænkel set theory. The main problem is due to the axiom of extensionality. To overcome this difficulty, we first give the axioms of ZF in a suitable (equivalent) form, which we call ZF". 1 The ZF" set theory This theory is written in the first order predicate calculus without equality, with only three binary relation symbols : 2, ⊂ (which have their usual meaning), and " (which is a kind of \ strong membership " relation). The formula x = y is an abbreviation for x ⊂ y ^ y ⊂ x. We shall use the notation (8x " a)F (x) for 8x(x " a ! F (x)), and (9x " a)F (x) for 9x(x " a ^ F (x)). The axioms are the following : 1 0. Equality and extensionality axioms. 8x8y[x 2 y $ (9z " y)x = z] ; 8x8y[x ⊂ y $ (8z " x)z 2 y]. 1. Foundation scheme. 8a[(8x " a)F (x) ! F (a)] ! 8a F (a) (for every formula F (x; x1; : : : ; xn)). The intuitive meaning of axioms 0 and 1 is that " is a well founded relation, and that the relation 2 is obtained by \ collapsing " " into an extensional binary relation. The following axioms essentially express that the relation " satisfies the axioms of Zermelo- Frænkel except extensionality. 2. Comprehension scheme. 8a9b8x[x " b $ (x " a ^ F (x))] (for every formula F (x; x1; : : : ; xn)). 3. Pairing axiom. 8a8b9x[a " x ^ b " x] 4. Union axiom. 8a9b(8x " a)(8y " x) y " b. 5. Power set scheme. 8a9b8x(9y " b)8z(z " y $ (z " a ^ F (z; x))) (for every formula F (z; x; x1; : : : ; xn)). 6. Collection scheme. 8a9b(8x " a)[9y F (x; y) ! (9y " b)F (x; y)] (for every formula F (x; y; x1; : : : ; xn)). 7. Infinity scheme. 8a9bfa " b ^ (8x " b)[9y F (x; y) ! (9y " b)F (x; y)]g (for every formula F (x; y; x1; : : : ; xn)). Remark. These axioms are clearly very redundant : indeed, the power set scheme contains the comprehension scheme, and the collection scheme could easily be merged in the infinity scheme. We give the axioms in this manner only in order to show the relation with ZF . Let us show that this theory is a conservative extension of ZF + AF (AF is the axiom of foundation : 8a(9x 2 a)(8y 2 x) y 2= a). In the first place, it is clear that, if ZF " ` F , where F is a formula of ZF (i.e. written only with 2 et ⊂), then ZF + AF ` F ; indeed, it is sufficient to notice that, if we replace " by 2 in ZF", we obtain a theory equivalent to ZF + AF . Conversely, we must show that each axiom of ZF + AF is a consequence of ZF " . Theorem 1. ZF " ` a ⊂ a (and thus a = a). We use the foundation scheme (this method is called \ induction on the rank of a "). We assume 8x(x " a ! x ⊂ x), and we must show a ⊂ a ; therefore, we add the hypothesis x " a. It follows that x ⊂ x, then x = x, and therefore 9y(x = y ^ y " a), that is to say x 2 a. Thus, we have 8x(x " a ! x 2 a), and therefore a ⊂ a. q.e.d. 2 Lemma 2. ZF " ` a ⊂ b; 8x(x 2 b ! x 2 c) ! a ⊂ c. We must show x " a ! x 2 c, which follows from x " a ! x 2 b and x 2 b ! x 2 c. q.e.d. If we replace a with b in lemma 2, we get Corollary 3. ZF " ` 8x(x 2 b ! x 2 c) ! b ⊂ c. Therefore, we have proved, in ZF " , the first axiom of ZF , namely : • Extensionality axiom : 8x8y(8z(z 2 x $ z 2 y) ! x = y)). Theorem 4. ZF " ` 8y8z(y = a; a 2 z ! y 2 z) ; 8y8z(a ⊂ y; z 2 a ! z 2 y). Call F (a), F 0(a) these two formulas. We show F (a) by induction on the rank of a. Thus, we suppose 8x(x " a ! F (x)). We first show F 0(a) : by hypothesis, we have a ⊂ y, z 2 a ; thus, there exists a0 such that z = a0 and a0 " a, and thus F (a0). From a0 " a and a ⊂ y, we deduce a0 2 y. From z = a0 and a0 2 y, we deduce z 2 y by F (a0). Then, we show F (a) : by hypothesis, we have y = a, a 2 z, thus a = y0 and y0 " z for some y0. In order to show y 2 z, it is sufficient to show y = y0. Now, we have y = a, a = y0, and thus y0 ⊂ a, a ⊂ y. From F 0(a), we get 8z(z 2 a ! z 2 y) ; from y0 ⊂ a, we deduce y0 ⊂ y by lemma 2. We have also y ⊂ a, a ⊂ y0. From F 0(a), we get 8z(z 2 a ! z 2 y0) ; from y ⊂ a, we deduce y ⊂ y0 by lemma 2. q.e.d. With corollary 3, we obtain : Corollary 5. ZF " ` b ⊂ c $ 8x(x 2 b ! x 2 c). It is now easy to deduce the equality axioms of ZF , namely : • Equality axioms : 8x(x = x), 8x8y(x = y ! y = x), 8x8y8z(x = y; y = z ! x = z), 8x8y8x08y0(x = x0; y = y0 ! (x ⊂ y $ x0 ⊂ y0) ^ (x 2 y $ x0 2 y0)). Remark. The equality = is an equivalence relation, which is compatible with the relations 2 et ⊂ but not with the relation ". • Foundation axiom : as is well known, it is equivalent to the scheme 8a[8x(x 2 a ! F (x)) ! F (a)] ! 8a F (a) (for every formula F (x; x1; : : : ; xn) which is written only with 2 and ⊂). From axiom scheme 1, it is sufficient to show : [8x(x 2 a ! F (x)) ! F (a)] ! [8x(x " a ! F (x)) ! F (a)], or else 8x(x " a ! F (x)) ! 8x(x 2 a ! F (x)). From x 2 a, we deduce x = x0 and x0 " a for some x0, thus F (x0), and finally F (x), since F is compatible with =. • Comprehension scheme : 8a9b8x[x 2 b $ (x 2 a ^ F (x))] (for every formula F (x; x1; : : : ; xn) written with ⊂, 2). From the axiom scheme 2 (comprehension scheme), we get 8x[x " b $ (x " a ^ F (x))]. If x 2 b, then x = x0, x0 " b for some x0. Thus x0 " a and F (x0). From x = x0 and x0 " a, we 3 deduce x 2 a. Since the axioms of equality are satisfied for ⊂ and 2, and therefore for F , we obtain F (x). Conversely, if we have F (x) and x 2 a, we have x = x0 and x0 " a for some x0. Since F is compatible with equality, we get F (x0), thus x0 " b and x 2 b. • Pairing axiom : 8x8y9z[x 2 z ^ y 2 z]. Trivial consequence of axiom 3, and Lemma 6. ZF" ` x " y ! x 2 y. Trivial consequence of axioms 0 and x = x. q.e.d. • Union axiom : 8a9b8x8y[x 2 a ^ y 2 x ! y 2 b]. From x 2 a we have x = x0 and x0 " a for some x0 ; we have y 2 x, therefore y 2 x0, thus y = y0 and y0 " x0. From axiom 4, x0 " a and y0 " x0, we get y0 " b ; therefore y 2 b, by y = y0. • Power set scheme : 8a9b8x9y[y 2 b ^ 8z(z 2 y $ (z 2 a ^ F (z; x)))] (for every formula F (z; x; x1; : : : ; xn) written with 2 and ⊂). From axiom scheme 5, we have y " b, and thus y 2 b. If z 2 y, we have z = z0 and z0 " y for some z0, therefore z0 " a and F (z0; x), thus z 2 a and F (z; x) (because F is compatible with =). Conversely, if z 2 a and F (z; x), we have z = z0 and z0 " a for some z0, therefore F (z0; x), thus z0 " y and z 2 y.