Embedding sheaf models for set theory into boolean-valued permutation models with an interior operator

Annals of Pure and Applied Logic 32 (1986) 103-109 North-Holland

103

E M B E D D I N G S H E A F MODELS FOR SET T H E O R Y INTO B O O L E A N - V A L U E D P E R M U T A T I O N MODELS WITH A N INTERIOR O P E R A T O R Andre SCEDROV Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA Communicated by A. Nerode Received 20 February 1985

Introduction Myhill [14] proposed a system of set theory based on $4 modal logic, which crystallized into its present form in [9, 10, 16, 17, 3]. In this setting, one works in classical logic, but at the same time one can make distinctions usually associated with intuitionism. A constructive existential quantifier, for example, can be expressed as ( : I x ) E l . . . . Epistemic ZF set theory is a conservative extension of both classical ZF set theory [9, 17] and intuitionistic ZF set theory [3] (the latter via an extension of a G6del-type modal interpretation of intuitionistic logic [15] to set theory [16]). Furthermore, epistemic ZF (like intuitionistic ZF, but unlike classical ZF) has the numerical existence property: if it proves a sentence :Ix • to ElA(x), then there is a (definable) standard numeral n such that A(n) is provable [9, 17]. Considering the slash interpretations [14, 9, 17] of epistemic ZF only as tools of technical nature, one wants to know whether this theory has any interesting models. By analogy with the Scott-Solovay Boolean-valued models of classical ZF [1], models over complete topological Boolean algebras [15] were proposed by Goodman [10] and used recently by Flagg [3]. On the other hand, intuitionistic ZF has a rich variety of recursion-theoretic, topological and category-theoretic interpretations. Here we concentrate on the Fourman interpretation [4] in Grothendieck toposes. We show that every instance of this interpretation can be extended to an interpretation of epistemic ZF given in terms of permutations on a topological Boolean algebra, very much akin to the Scott-Solovay symmetric extensions in classical ZF. We also show that this embedding is an algebraic counterpart to the syntactic translation given in [16]. We use a powerful representation theorem of Freyd [7, 4§7, 13] which reduces the problem to the consideration of the Fourman interpretation in Heyting-valued extensions of a particular permutation model. The other crucial ingredient is an embedding of complete Heyting algebras into complete Boolean algebras [6§§2.20,6.15], 0168-0072/86/$3.50 (~) 1986, Elsevier Science Publishers B.V. (North-Holland)

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apparently first obtained by Funayama [12, §7.55] by different methods. The relevance of this embedding for epistemic systems was first made clear by Flagg [2] (w.r.t. epistemic type theory). Our results were first obtained in September 1983. We were motivated by our earlier joint work with Mike Fourman on the verifications of the Funayama embedding for elementary toposes. Both of these were briefly announced in [5], together with additional observations of P. Freyd. It is a pleasure to acknowledge their interaction with us.

1. Epistemic set theory The language of epistemic set theory Z F E is the first-order language with equality, and a binary relation symbol e. The logical symbols are _1., v , ^, ---~, :!, V, D. Z F E has the following axioms and rules: (a) Underlying logic

(o) (1) (2) (3) (4) (5) (6) (7) (8) (9) (lO)

Equality axioms, x = y ^ A (x) --> A (y). All classical propositional tautologies. From A and A---> B infer B, C1A---,A. DA---> DDA. [24 ^ FI(A---> B)"-* F1B. From A infer EtA. VxA(x)--->A(y), where y is free for x in A(x). From A-->B(x) infer A--->VxB(x), i f x is not free in A. A(y)--->:ixA(x), where y is free for x i n A ( x ) . From B(x)--->A infer 3x B(x)--->A, if x is not free in A.

(b) Non-logical axioms

Epistemic Extensionality: DVz (z ~ x ---, z e y) --->x = y. Foundation: Vx (Vy e x A(y)---> A(x))---> Vx A(x). Epistemic Foundation: ElVx (l'-lVy ¢ x A (y) --->A (x)) ~ DVx A (x). Pairing: 3z E](x ~ z A y ~ Z ). (15) Union: 3z rqVw (3y ~ x w e y ---, w e z). (16) Separation: :lz DVy (y ¢ z *->y ¢ x h A ( y ) ) , where z is not free in A(y), (17) Epistemic Power Set: 3z DVw (DVy ~ w y ~ x---> w ~ z), (18) Infinity: 3z D(:ly Cl(y ~ z) ^ Vu ~ z =Iv D(v ~ z ^ u ~ v)), (19) Collection: Vx ~ u 3y A(x, y)---> 3z Vx ~ u 3y ~ z A(x, y), where z is not free in a(x, y), (20) Epistemic Collection: rTVx ~ u :ly A(x, y)---> 3z DVx ~ u 3y (D(y ~ z) ^ A(x, y)), where z is not free in A (x, y). (11) (12) (13) (14)

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We recall that intuitionistic set theory ZFI is based on Heyting's predicate calculus with equality, and the nonlogical axioms are obtained by erasing all modalities in (11)-(20). We defined a syntactic translation (.)* from ZFI to ZFE as follows [16]: Let HS(x) be a formula in the language of ZFE saying that "x is hereditarily []-stable", i.e., H S ( x ) =- 3 u D ( x e u A V y e u V z e y z e u ^ V y e u V z e y D(z ey)).

Now let: _b * ~

_L,

(x ~ y)* =---x ~ y, (x = y ) * - x

=y,

(A ^ B ) * = - A * A B*, (A v B ) * = - A * v B*, (A--> B )* - D ( A * --~ B* ),

(Vx A ) * --- I-]Vx (HS(x) --> A *), (:Ix A) * - 3x (HS(x) ^ A *). We showed in [16] that for any sentence A provable in ZFI, A* is provable in ZFE. The converse was recently shown by Flagg [3].

2. Boolean-valued permutation models of ZFE There are several equivalent approaches to Boolean-valued permutation models (i.e. symmetric extensions) for ZF, e.g. [1, chapter 3] or [4, §§3-5]. Our development of similar models for Z F E will be closer to the latter approach. Let B be a Boolean algebra equipped with an interior operator, i.e. a map [] : B ~ B such that (i) D1 = 1, (ii) D(p ^ q) = Dp ^ • q , for any p, q e B, (iii) • p DDp = • p , for any p e B. (These were called topological Boolean algebras in [15].) Let G be a topological group acting continuously on B, with B thought of as a discrete space (i.e. the stabilizer of each p e B is an open subgroup of G). We assume in addition that for each open subgroup H, each H-stable subset of B has an H-stable sup (in0 in B. Furthermore, we assume that [ ] : B ~ B is a G-equivariant map, i.e., g . Op = D(g . p )

for e a c h g e G, p e B.

We define the sets V~c's~ by transfinite recursion on an ordinal tr as follows. Let V~6,n~ be the set of all functions x with an open stabilizer and with

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rge(x) c_ B, and dom(x) ~_ V~6,m for some fl < o~. Here g . x is defined by transfinite recursion as { (g- y, g - (x(y))): y e dom(x)}. Let V
IIa(xl,...,

xn)ll ~ n as fonows:

II ± II = 0, IIx ~ yll

=

V{y(z)

^

IIx-yll = • A { x ( z ) ~ ^

IIx = zll: z ~ dora(y)}, IIz ~yll: z ~ dom(x)}

[] A {y (z ) ---> IIz ~xll: z ~ dom(y)},

IIA1 ^ a ~ l l - IIAill ^ IIA~ll, IIA1 v A ~ l l - IIA~ll v IIA~ll, IIA1--->A=ll = IIaill--, IIa~ll, 113xA(x)ll = A{IIA(u)II: u ~ v(~,B)},

IlVx A(x)ll = A(IIA(u)II: u ~ v(~.')}, I 1 ~ 1 1 - [] IIAII. Heyting-valued permutation models of ZFI are defined analogously, disregarding [] and letting B be a Heyting algebra [4, 11, 18]. Theorem 2.1. S u p p o s e ZFE p r o v e s A ( Z l , . . . , z , ) , w h e r e all f r e e variables are as exhibited. T h e n f o r each x l , . . . , x , ~ V f3 V ~6"m, z ( w ) = 1 for every w e dom(z). Clearly, z is stabilized by all of G. We show that for all w e V (6'm, []llw :_xll ~< IIw ~ zll. Indeed, for every w ~ V ~6"B> let w' be given by dom(w') = dom(x), w ' ( y ) = Ily e wll. Note that w' e dom(z). It is readily shown that IIw':_ wll = 1 and IIw n x ~_ w'll = 1,

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and therefore DIIw xll IIw = w'l, so clearly DIIw xll IIw zll. Note that the formulation of this axiom using "[](w _ x)" rather than just "w c_x" is used critically in the next to the last step.

3. Embedding sheaf models into Boolean-valued permutation models In [4], Fourman gave an interpretation of ZFI in any Grothendieck topos, thus showing that these sheaf categories, which were first developed as an essential conceptual tool in the study of algebraic geometry and algebraic topology, are intrinsically connected with intutionistic set theory. Furthermore, one has the following representation theorem [7, 4§7, 13]: Theorem 3.1 (Freyd). Let Go be the topological group of permutations o f a countably enumerated set A, so that a subgroup H <- Go is open iff it contains a subgroup that pointwise stabilizes some finite initial segment o f A. Let ~g be any Grothendieck topos. Then there is a Heyting-valued permutation model V B definable in terms o f the cHa L. Proposition 3.3. Let f: L ~ B be as in Theorem 3.2. Let E]" B--->B be defined as I"lb = V {f(p): f ( p ) <-b, p ~ L}. Then (B, D) is a complete topological Boolean algebra, and f(p---> q) = E](f(p)--> f(q)), and f ( A i P i ) = I'q/~if(Pi) for all p, q, Pi ~ L. Furthermore, b = Fqb iff b = f ( p ) for some p ¢. L. ProoL Straightforward, recalling that p---> q = V{r [ r ^ p ~
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embedding f: L ,-->B. (We can think of V (a°'L) as the part of V (c'°'m consisting of hereditarily L-valued sets.) Recalling the syntactic translation (.)* from ZFI to Z F E mentioned in Section 1, we state our main result: Theorem 3.4. Let Go be the topological group given in Theorem 3.1. Let ~g be a Grothendieck topos. Then there exists a Boolean-valued permutation model V ~°°'n) o f epistemic set theory ZFE so that f o r every sentence o f ZFI, A is valid under the Fourman interpretation in ~g iff I l a * l l - 1 in V (c~'a).

We will need the following lemmas: °

Lemma 3.5. For every x • V (c'°'m, IIHS(x)II-- V {llx = x'll x' • v<~,'>} Proof. We consider V (c'°'z) as part of V (c'°'m, considering f as the inclusion. It is readily checked by transfinite induction on the rank of x ' e V (c'°'L) that IlHS(x')l[ = 1. By Extensionality in V (c'°'m, it thus suffices to show that for every x • V (c'°'B), IIHS(x)ll ~ V {llx = x ' l l x ' • v<~,'->}.

We establish this by showing that for every x, u • V (~'8) there exists x' • V (~'z')

such that DIIx e u ^ Vv • u V w e v w e u ^ Vo e u V w • o D ( w e ~)11 <~ IIx = x'll.

We establish this property of x by transfinite induction on the rank of x • V (c'°'B). Let p be the left-hand side of the inequality, for a given x, u • V (6°'B). By the induction hypothesis, the property holds for every y • dora(x). In particular, for this chosen u, p ^ Ily ~xll ~< Ily ~ull. Thus there exists y ' e V (GO'L) such that P ^ Ily • xll -

[]

Lemma 3.6. Let A ( x ) be a formula o f ZFI with precisely x free. Then in V (c°'m,

II(Vx a(x))*ll = D A {llA*(x')ll" x' • v<~,')},

and

11(3xA(x))*ll- V {llA*(x')ll:x' e V<~,L)}. ProoL Lemma 3.5, and Extensionality. Proof of T h e o r e m 3.4. Lemma 3.6, and Proposition 3.3.

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We conclude with a few remarks. If L is Boolean, then B = L, and [] = identity. In particular, choose B so that the Axiom of Choice fails in V (c'°m) considered as a Boolean-valued permutation model of ZF. Therefore, the embedding considered here differs from Barr's topos-theoretic construction [12, Chapter 7]. Granting that the Fourman interpretation [4] can be made constructive, one could work entirely in ~, and take L = ~(1). Such an approach would be similar to [3], where one worked provably in ZFI, but applying furthermore the Fourman interpretation in ~. Because the real world ordinals are cofinal in ~, one would obtain a Boolean-valued model satisfying the same sentences of ZFE as the model obtained by Theorem 3.4.

References [1] J.L. Bell, Boolean-valued Models and Independence Proofs in Set Theory (Oxford University Press, Oxford, 1977). [2] R.C. Flagg, Integrating classical and intuitionistic mathematics, Ann. Pure Appl. Logic, to appear. [3] R.C. Flagg, Epistemic set theory is a conservative extension of intuitionistic set theory, J. Symbolic Logic 50 (1985) 895-902. [4] M.P. Fourman, Sheaf models for set theory, J. Pure Appl. Algebra 19 (1980) 91-101. [5] M.P. Fourman, P. Freyd, A. S~edrov, Toposes and modal logic, Abstracts of the 1983-84 A.S.L. Annual Meeting, J. Symbolic Logic 49 (1984) 1443. [6] M.P. Fourman, D.S. Scott, Sheaves and logic, in: M.P. Fourman, C.J. Mulvey, and D.S.Scott, eds., Applications of Sheaves, Lecture Notes in Math. 753 (Springer, Berlin, 1979) 302-401. [7] P. Freyd, All topoi are localic, or: Why permutation models prevail, J. Pure Appl. Algebra, to appear. [8] K. G6del, Eine Interpretation des Intuitionistichen Aussagenkalkuls, Ergebnisse eines mathematischen Kolloquiums 4 (1932) 39-40. [9] N.D. Goodman, A genuinely intensional set theory, in: S. Shapiro, ed., Intensional Mathematics (North-Holland, Amsterdam, 1985). [10] N.D. Goodman, Boolean-valued models of epistemic set theory, Preprint, 1983. [11] R. Grayson, Heyting-valued models for intttitionistic set theory, in: M.P. Fourman, C.J. Mulvey, and D.S. Scott, eds., Applications of Sheaves, Lecture Notes in Math. 753 (Springer, Berlin, 1979) 402-414. [12] P.T. Johnstone, Topos Theory (Academic Press, London, 1977). [13] P.T. Johnstone, Quotients of decidable objects in a topos, Math. Proc. Cambridge Philos. Soc. 93 (1983) 409-419. [14] J. Myhill, Intensional set theory, in: S. Shapiro, ed., Intensional Mathematics (North-Holland, Amsterdam, 1985). [15] H. Rasiowa and R. Sikorski, The Mathematics of Metamathematics (North-Holland, Amsterdam, 1963). [16] A. S~edrov, Extending G6del's modal interpretation to type theory and set theory, in: S. Shal~iro, Intensional Mathematics (North-Holland, Amsterdam, 1985). [17] A. S~edrov, Some properties of epistemic set theory with collection, J. Symbolic Logic, to appear. [18] A. ~edrov, Consistency and independence results in intuitionistic set theory, in; F. Riehman, ed., Constructive Mathematics, Lecture Notes in Math. 873 (Springer, Berlin, 1981, 54-86.