Intuitionistic choice and classical logic - Semantic Scholar
Intuitionistic choice and classical logic
Thierry Coquand Erik Palmgreny February 27, 1997
1. Introduction
The e ort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [2]. In this paper we show how to combine the unrestricted countable choice, induction on in nite well-founded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand the extensional version of HA! with a new basic type for the rst tree class. Add countable and dependent choice together with the axiom of unique choice at all types. The equality relations on the two basic types are supposed to be decidable. Moreover we assume the following numerical omniscience scheme (8n) [(x; n) _ :(x; n)] ?! (8n) (x; n) _ (9n) :(x; n); where is any formula and n ranges over the natural numbers. It may be argued that for the development of mathematics it is more natural to limit the law of excluded middle than the choice principles. A theorem, well known in topos theory, states that the axiom of choice (and hence the principle of excluded middle) is valid in a topos E if, and only if, E is equivalent to the topos of sheaves over a complete boolean algebra (see [7]). This theorem is non-constructive and relies on Zorn's lemma. In this paper we modify the idea of this model in order to give a constructive model of the above mentioned theory. This is done by considering boolean algebras that are merely -complete and whose topologies are generated by nite or countable covering relations. By a judicious choice of the boolean algebra we can directly extract e ective content from 2 -statements true in the model. The use of boolean models to give constructive interpretation of non-constructive principles was initiated by Coquand [3]. All the arguments of the present paper can be formalised in Martin-Lof's constructive type theory. A short outline of the paper follows. In Section 2 we show how to construct -complete boolean algebras with compact and non-compact topologies. The sheaves over these topologies are studied. Various choice principles and the extraction property for 2 -statements are derived in Section 3. Section 4 examines the extent to which classical logic is valid in the model. It is shown that decidable formulae are closed under quanti cation over natural E-mail: [email protected] The second author's research was supported by the Swedish Natural Science Research Council (NFR). E-mail: [email protected] y
1
numbers. In Section 5, the model is extended to in nite well-founded trees (the rst tree class [2]). The special features of the model over the compact topology are scrutinised in Section 6. Finally, Section 7 presents some applications of the model. A semi-constructive set theory is interpreted, and an example of a typical non-constructive argument yielding an e ective 02-theorem is given.
2. Sheaves over Boolean Algebras
We rst present a quite general construction of a -complete boolean algebra which permits extraction of e ective content from true 02 -propositions. It appeared in [11] and could be considered as a semantic version of the Lindenbaum construction. A proof-theoretic construction is given in Martin-Lof [10] of the -complete boolean algebra of Borel subsets of the Cantor space. Let F be a xed proposition. Let H (x) be a family of propositions over the set G. De ne a new set B together with a new family of propositions C (p) over B inductively as follows: (i) 0; 1 2 B and C (0) = F and C (1) = >, (ii) for any x 2 G, let xg 2 B and C (xg ) = (H (x) ! F ) ! F , (iii) if p 2 B , then :p 2 B and C (:p) = C (p) ! F , V (iv) if pn 2 B for all n 2 N , then n pn 2 B and ^ C ( pn) = (n 2 N ) C (pn):
V
n
Note that :, and g are so far only regarded as constructors. To this inductive generation V corresponds naturally an induction principle. De ne p ^ q = n rn , where r0 = p and rn+1 = q. We often simply write pq for p ^ q. Moreover, let p _ q = :(:p ^ :q), Wn pn = : Vn :pn. In a manner similar to the Lindenbaum algebra construction we introduce an equality on B : p q i C (p) $ C (q):
Theorem 1 For any proposition F and any basic family H (x) over G,Vthe Wset B de ned above with equality is a -complete boolean algebra with operations ^; ; _; :. Furthermore if F = (9n 2 N )H (f (n)), then the extraction property ^ C (: :f (n)g ) ) F n
holds.
Proof. By induction on B one proves that for all p ::p p: The veri cation of the axioms for a boolean algebra is then straightforward by intuitionistic logic. The second part follows also by purely logical reasoning. As two important examples we have the case when G is an empty set (no basic propositions) and the case when (G; H ) is the universe of small sets (U; T ). We shall use the latter 2
in the sequel. Let S be a small set, with a small equality =S , i.e. it is given by a (propositional) function from S S into the universe of small sets. Then the boolean valued equality, [x y ] = (x =S y )g , satis es the following conditions: [x x] = 1, [x y ] = [y x] and [x y ] ^ [y z ] [x z ]. Let B be a -complete boolean algebra as above. Regard it as a category in the usual way. We shall impose two generalised topologies on it, one compact K and one non-compact J . First we introduce some notation. Let Nk = f0; 1; : : :; k ? 1g be the canonical k-element set, and write N! for N . A nite or in nite sequence of elements in B is then a function p : Nm ! B where m = 0; 1; 2; : : :; !. It is usually written as (pk )k<m .
De nition 2 Let (pk )k<m be a sequence of elements in B of length m !. This sequence is called a J -cover of p 2 B if W p = p; (i) k<m k
(ii) pj pk = 0 for all j < k < m. The notion of K -cover is similar except that only nite sequences can be covers. We shall use partition as a synonym for cover. It may now be checked that this notion satis es the usual requirements for a basic cover that generate a Grothendieck topology [9]. The only nontrivial part is composition of J covers. Suppose that (rij )j<mi J -covers qi for each i < m, and that (qi )i<m J -covers p. We let the new cover be (pk )k= k according to Cantor's pairing function, and put pk = 0 otherwise. In the sequel we will not bother with explicit coding of pairs.
Remark 3 (B; K ) is the usual base which gives a Stone space [8]. Let L be either of the topologies J or K . Note that it is required of a sheaf F : B op ! Set over (B; L) only that it be a pre-sheaf, and that it satis es the condition that if (pk )k<m is a partition of p and xk 2 F (pk ) for k < m, then there is a unique x 2 F (p) such that xk = Fpk ;p(x) for all k < m. The usual compatibility requirement vanishes since F (0) is a singleton. For any small set S with small equality =S we now de ne the sheaf of locally constant S -objects S^L : B op ! Set. Writing S^ = S^L, let the elements of S^(p) be pairs consisting of an L-cover (pk )k<m of p and a sequence (sk )k<m of objects in S . The intuitive interpretation of this pair of sequences is as a function taking the part pk to sk . We denote the elements by hpk 7! sk ik<m, and write simply hp0 7! s0i if m = 1. Two such elements hpk 7! sk ik<m and hqj 7! tj ij
(E2) E (sup(f ); 0) is false (E3) E (0; sup(g )) is false (E4) E (sup(f ); sup(g )) i 8m 9n E (f (m); g(n)) and 8n 9m E (f (m); g (n)). The formula E can be constructed by rst deducing trans nite recursion, which follows from trans nite induction (5) and the axiom of unique choice. This axiom also yields characteristic functions for decidable formulae, and functionals expressing universal and existential quanti cation. We will not go into details here. By induction on trees it follows that any E satisfying (E1) { (E4) is an equivalence relation and is decidable. We de ne the basic set-theoretic relations on O as follows
x = y ,def E (x; y); x 2 y ,def 9z 1 y [x = z]: It is easily checked that extensionality is valid:
x = y i 8z [z 2 x , z 2 y]:
(10)
A set-theoretic formula is built up from = and 2 as basic relations, using logical connectives and quanti cation over O. Such a formula is called restricted if all of its quanti ers occur as (8x 2 O)[x 2 y ! ] or as (9x 2 O)[x 2 y ^ ]. We use the abbreviations (8x 2 y ) and (9x 2 y ) , respectively, for these bounded quanti cations. Note that for a set-theoretic formula
(8x 2 y )(u; x) , (8x 1 y )(u; x); (9x 2 y )(u; x) , (9x 1 y )(u; x): Hence by Lemma 23 we have Lemma 25 (REM) The principle of excluded middle holds for any restricted formula. Let CZF? be the system of constructive set theory presented by Aczel [1], but lacking the subset collection axiom.
Theorem 26 The axioms of CZF? +REM are true on the interpretations of 2 and = given above.
Proof. We verify here only a few crucial axioms. By (10), extensionality is clear. REM
is the lemma above. The set induction axiom follows directly from induction on trees (5) and the interpretation of restricted quanti cation. The axiom of strong collection is veri ed using countable choice. As for the axiom of restricted separation, suppose that (u; y ) is a restricted formula. We have to nd a set z such that
(8y ) [y 2 z , y 2 x ^ (u; y )]: 14
(11)
If x = 0, then let z = 0. Suppose now x = sup(f ). In case (u; f (n)) is false for all n, put z = 0. Otherwise, let n0 be some number for which it is true. Then de ne g : N ! O by letting g (m) = f (m) if (u; f (m)), and letting g (m) = f (n0 ) otherwise (this is possible by the axiom of unique choice). Thus z = sup(g ) satis es (11).
Remarks 27 (i) The theory CZF+REM presented in [1] is in fact impredicative. This can be seen by the following argument. For any set A we may form B = f0; 1gA using
exponentiation (i.e. subset collection). Then any restricted formula (x), which may contain quanti ers over B , de nes a set f = f(x; y) 2 A f0; 1g : (x) ^ y = 1 _ :(x) ^ y = 0g by an application of restricted separation. Since any restricted formula is decidable, f is a function in B . Clearly f (x) = 1 i (x). (ii) Moreover, extending CZF? with full classical logic, leads to the full separation scheme for sets of natural numbers. This is also an impredicative principle. Let (x) be any formula. Then (8x 2 ! ) (9z ) [(x) ^ z = hx; 1i _ :(x) ^ z = hx; 0i]: If we let (x; z ) denote the expression within square brackets, we have by strong collection some set d such that (8x 2 ! ) (9z 2 d) (x; z ) ^ (8z 2 d) (9x 2 ! ) (x; z ). By separation we nd u such that x 2 u i x 2 ! and hx; 1i 2 d. That is, u is the set of numbers x such that (x). (iii) The theory CZF? has in fact the same proof-theoretic strength as CZF, as was noted in [6].
7.2. Fully Classical Theories
We note that the model makes it possible to use a double negation translation to give a constructive interpretation of full classical logic together with instances of countable choice. This follows easily since 8m ::9n (m; n) is equivalent to 8m 9n (m; n) in the model for decidable . Regrettably, the naive interpretation of Kripke{Platek set theory into CZF? + REM fails, because of the collection axiom.
7.3. Combinatorial Theorems We present now a typical non e ective reasoning that can be interpreted in our model. Proposition 28 Let (un) be a sequence of integers. There exists an in nite sequence n0 < n1 < n2 < : : : such that un0 un1 un2 : : : Proof. By the principle of excluded middle for arithmetical formulae, the following formula holds for any k (9n > k)(8m > k) un um and hence we have (8k)(9n > k)(8m > k) un um : The claim follows by using dependent choices. A possible corollary which has a form of an existential statement is: 15
Corollary 29 Let (un) and (vn) be two sequences of integers. There exist p < q such that both up uq and vp vq : Proof. Using twice the proposition, we can nd n0 < n1 < : : : such that both extracted
sequences (unk ) and (vnk ) are increasing. We can then take p = n0 and q = n1 . This reasoning can be interpreted in our model. It follows that the following statement is constructively valid (8(un ); (vn))(9p < q )[up uq ^ vp vq ]: In the same way, it would be possible to show in our model the in nite version of Ramsey's theorem [5]. Since the proof-theoretic strength of the theory that we are able interpret is at least the Howard ordinal, we should expect stronger combinatorial principles to be true in the model. But the investigation of this is beyond the scope of this paper.
References
[1] P. Aczel. The type theoretic interpretation of constructive set theory. In: A. Macintyre, L. Pacholski, J. Paris, eds., Logic Colloquium '77, North-Holland 1978. [2] W. Buchholz, S. Feferman, W. Pohlers, W. Sieg, eds. Iterated Inductive De nitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Springer Lecture Notes in Mathematics, Vol. 897. Springer 1981. [3] T. Coquand. A formal space of ultra lters, manuscript. [4] S. Feferman. Theories of nite type related to mathematicalpractice. In: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland 1977, 913 { 971. [5] R.L. Graham, B.L. Rothschild and J.H. Spencer. Ramsey Theory, 2nd ed. Wiley-Interscience 1990. [6] E.R. Gri or and M. Rathjen. The strength of some Martin-Lof type theories. Archive for Mathematical Logic 33(1994), 347 { 385. [7] P.T. Johnstone. Topos Theory. Academic Press 1977. [8] P.T. Johnstone. Stone Spaces. Cambridge University Press 1982. [9] S. Mac Lane and I. Moerdijk. Sheaves in Geometry and Logic. Springer 1991. [10] P. Martin-Lof. Notes on Constructive Mathematics. Almkvist och Wiksell 1970. [11] E. Palmgren. On universes in type theory. U.U.D.M. Report 1996:19. [12] E. Palmgren. Sheaf-theoretic nonstandard analysis: constructive aspects. U.U.D.M. Report 1996:28. [13] G. Takeuti. Two Applications of Logic to Mathematics. Iwanami Shoten and Princeton University Press 1978. [14] A.S. Troelstra, ed. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer Lecture Notes in Mathematics, Vol. 344. Springer 1973. Thierry Coquand, Department of Computer Science Erik Palmgren, Department of Mathematics Chalmers University of Technology and Gothenburg University S-412 96 Gothenburg, SWEDEN
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Thierry Coquand Erik Palmgreny February 27, 1997
1. Introduction
The e ort in providing constructive and predicative meaning to non-constructive modes of reasoning has almost without exception been applied to theories with full classical logic [2]. In this paper we show how to combine the unrestricted countable choice, induction on in nite well-founded trees and restricted classical logic in a constructively given model. For readers faniliar with intuitionistic systems [14], we may succinctly describe the theory we interpret as follows. Expand the extensional version of HA! with a new basic type for the rst tree class. Add countable and dependent choice together with the axiom of unique choice at all types. The equality relations on the two basic types are supposed to be decidable. Moreover we assume the following numerical omniscience scheme (8n) [(x; n) _ :(x; n)] ?! (8n) (x; n) _ (9n) :(x; n); where is any formula and n ranges over the natural numbers. It may be argued that for the development of mathematics it is more natural to limit the law of excluded middle than the choice principles. A theorem, well known in topos theory, states that the axiom of choice (and hence the principle of excluded middle) is valid in a topos E if, and only if, E is equivalent to the topos of sheaves over a complete boolean algebra (see [7]). This theorem is non-constructive and relies on Zorn's lemma. In this paper we modify the idea of this model in order to give a constructive model of the above mentioned theory. This is done by considering boolean algebras that are merely -complete and whose topologies are generated by nite or countable covering relations. By a judicious choice of the boolean algebra we can directly extract e ective content from 2 -statements true in the model. The use of boolean models to give constructive interpretation of non-constructive principles was initiated by Coquand [3]. All the arguments of the present paper can be formalised in Martin-Lof's constructive type theory. A short outline of the paper follows. In Section 2 we show how to construct -complete boolean algebras with compact and non-compact topologies. The sheaves over these topologies are studied. Various choice principles and the extraction property for 2 -statements are derived in Section 3. Section 4 examines the extent to which classical logic is valid in the model. It is shown that decidable formulae are closed under quanti cation over natural E-mail: [email protected] The second author's research was supported by the Swedish Natural Science Research Council (NFR). E-mail: [email protected] y
1
numbers. In Section 5, the model is extended to in nite well-founded trees (the rst tree class [2]). The special features of the model over the compact topology are scrutinised in Section 6. Finally, Section 7 presents some applications of the model. A semi-constructive set theory is interpreted, and an example of a typical non-constructive argument yielding an e ective 02-theorem is given.
2. Sheaves over Boolean Algebras
We rst present a quite general construction of a -complete boolean algebra which permits extraction of e ective content from true 02 -propositions. It appeared in [11] and could be considered as a semantic version of the Lindenbaum construction. A proof-theoretic construction is given in Martin-Lof [10] of the -complete boolean algebra of Borel subsets of the Cantor space. Let F be a xed proposition. Let H (x) be a family of propositions over the set G. De ne a new set B together with a new family of propositions C (p) over B inductively as follows: (i) 0; 1 2 B and C (0) = F and C (1) = >, (ii) for any x 2 G, let xg 2 B and C (xg ) = (H (x) ! F ) ! F , (iii) if p 2 B , then :p 2 B and C (:p) = C (p) ! F , V (iv) if pn 2 B for all n 2 N , then n pn 2 B and ^ C ( pn) = (n 2 N ) C (pn):
V
n
Note that :, and g are so far only regarded as constructors. To this inductive generation V corresponds naturally an induction principle. De ne p ^ q = n rn , where r0 = p and rn+1 = q. We often simply write pq for p ^ q. Moreover, let p _ q = :(:p ^ :q), Wn pn = : Vn :pn. In a manner similar to the Lindenbaum algebra construction we introduce an equality on B : p q i C (p) $ C (q):
Theorem 1 For any proposition F and any basic family H (x) over G,Vthe Wset B de ned above with equality is a -complete boolean algebra with operations ^; ; _; :. Furthermore if F = (9n 2 N )H (f (n)), then the extraction property ^ C (: :f (n)g ) ) F n
holds.
Proof. By induction on B one proves that for all p ::p p: The veri cation of the axioms for a boolean algebra is then straightforward by intuitionistic logic. The second part follows also by purely logical reasoning. As two important examples we have the case when G is an empty set (no basic propositions) and the case when (G; H ) is the universe of small sets (U; T ). We shall use the latter 2
in the sequel. Let S be a small set, with a small equality =S , i.e. it is given by a (propositional) function from S S into the universe of small sets. Then the boolean valued equality, [x y ] = (x =S y )g , satis es the following conditions: [x x] = 1, [x y ] = [y x] and [x y ] ^ [y z ] [x z ]. Let B be a -complete boolean algebra as above. Regard it as a category in the usual way. We shall impose two generalised topologies on it, one compact K and one non-compact J . First we introduce some notation. Let Nk = f0; 1; : : :; k ? 1g be the canonical k-element set, and write N! for N . A nite or in nite sequence of elements in B is then a function p : Nm ! B where m = 0; 1; 2; : : :; !. It is usually written as (pk )k<m .
De nition 2 Let (pk )k<m be a sequence of elements in B of length m !. This sequence is called a J -cover of p 2 B if W p = p; (i) k<m k
(ii) pj pk = 0 for all j < k < m. The notion of K -cover is similar except that only nite sequences can be covers. We shall use partition as a synonym for cover. It may now be checked that this notion satis es the usual requirements for a basic cover that generate a Grothendieck topology [9]. The only nontrivial part is composition of J covers. Suppose that (rij )j<mi J -covers qi for each i < m, and that (qi )i<m J -covers p. We let the new cover be (pk )k= k according to Cantor's pairing function, and put pk = 0 otherwise. In the sequel we will not bother with explicit coding of pairs.
Remark 3 (B; K ) is the usual base which gives a Stone space [8]. Let L be either of the topologies J or K . Note that it is required of a sheaf F : B op ! Set over (B; L) only that it be a pre-sheaf, and that it satis es the condition that if (pk )k<m is a partition of p and xk 2 F (pk ) for k < m, then there is a unique x 2 F (p) such that xk = Fpk ;p(x) for all k < m. The usual compatibility requirement vanishes since F (0) is a singleton. For any small set S with small equality =S we now de ne the sheaf of locally constant S -objects S^L : B op ! Set. Writing S^ = S^L, let the elements of S^(p) be pairs consisting of an L-cover (pk )k<m of p and a sequence (sk )k<m of objects in S . The intuitive interpretation of this pair of sequences is as a function taking the part pk to sk . We denote the elements by hpk 7! sk ik<m, and write simply hp0 7! s0i if m = 1. Two such elements hpk 7! sk ik<m and hqj 7! tj ij
(E2) E (sup(f ); 0) is false (E3) E (0; sup(g )) is false (E4) E (sup(f ); sup(g )) i 8m 9n E (f (m); g(n)) and 8n 9m E (f (m); g (n)). The formula E can be constructed by rst deducing trans nite recursion, which follows from trans nite induction (5) and the axiom of unique choice. This axiom also yields characteristic functions for decidable formulae, and functionals expressing universal and existential quanti cation. We will not go into details here. By induction on trees it follows that any E satisfying (E1) { (E4) is an equivalence relation and is decidable. We de ne the basic set-theoretic relations on O as follows
x = y ,def E (x; y); x 2 y ,def 9z 1 y [x = z]: It is easily checked that extensionality is valid:
x = y i 8z [z 2 x , z 2 y]:
(10)
A set-theoretic formula is built up from = and 2 as basic relations, using logical connectives and quanti cation over O. Such a formula is called restricted if all of its quanti ers occur as (8x 2 O)[x 2 y ! ] or as (9x 2 O)[x 2 y ^ ]. We use the abbreviations (8x 2 y ) and (9x 2 y ) , respectively, for these bounded quanti cations. Note that for a set-theoretic formula
(8x 2 y )(u; x) , (8x 1 y )(u; x); (9x 2 y )(u; x) , (9x 1 y )(u; x): Hence by Lemma 23 we have Lemma 25 (REM) The principle of excluded middle holds for any restricted formula. Let CZF? be the system of constructive set theory presented by Aczel [1], but lacking the subset collection axiom.
Theorem 26 The axioms of CZF? +REM are true on the interpretations of 2 and = given above.
Proof. We verify here only a few crucial axioms. By (10), extensionality is clear. REM
is the lemma above. The set induction axiom follows directly from induction on trees (5) and the interpretation of restricted quanti cation. The axiom of strong collection is veri ed using countable choice. As for the axiom of restricted separation, suppose that (u; y ) is a restricted formula. We have to nd a set z such that
(8y ) [y 2 z , y 2 x ^ (u; y )]: 14
(11)
If x = 0, then let z = 0. Suppose now x = sup(f ). In case (u; f (n)) is false for all n, put z = 0. Otherwise, let n0 be some number for which it is true. Then de ne g : N ! O by letting g (m) = f (m) if (u; f (m)), and letting g (m) = f (n0 ) otherwise (this is possible by the axiom of unique choice). Thus z = sup(g ) satis es (11).
Remarks 27 (i) The theory CZF+REM presented in [1] is in fact impredicative. This can be seen by the following argument. For any set A we may form B = f0; 1gA using
exponentiation (i.e. subset collection). Then any restricted formula (x), which may contain quanti ers over B , de nes a set f = f(x; y) 2 A f0; 1g : (x) ^ y = 1 _ :(x) ^ y = 0g by an application of restricted separation. Since any restricted formula is decidable, f is a function in B . Clearly f (x) = 1 i (x). (ii) Moreover, extending CZF? with full classical logic, leads to the full separation scheme for sets of natural numbers. This is also an impredicative principle. Let (x) be any formula. Then (8x 2 ! ) (9z ) [(x) ^ z = hx; 1i _ :(x) ^ z = hx; 0i]: If we let (x; z ) denote the expression within square brackets, we have by strong collection some set d such that (8x 2 ! ) (9z 2 d) (x; z ) ^ (8z 2 d) (9x 2 ! ) (x; z ). By separation we nd u such that x 2 u i x 2 ! and hx; 1i 2 d. That is, u is the set of numbers x such that (x). (iii) The theory CZF? has in fact the same proof-theoretic strength as CZF, as was noted in [6].
7.2. Fully Classical Theories
We note that the model makes it possible to use a double negation translation to give a constructive interpretation of full classical logic together with instances of countable choice. This follows easily since 8m ::9n (m; n) is equivalent to 8m 9n (m; n) in the model for decidable . Regrettably, the naive interpretation of Kripke{Platek set theory into CZF? + REM fails, because of the collection axiom.
7.3. Combinatorial Theorems We present now a typical non e ective reasoning that can be interpreted in our model. Proposition 28 Let (un) be a sequence of integers. There exists an in nite sequence n0 < n1 < n2 < : : : such that un0 un1 un2 : : : Proof. By the principle of excluded middle for arithmetical formulae, the following formula holds for any k (9n > k)(8m > k) un um and hence we have (8k)(9n > k)(8m > k) un um : The claim follows by using dependent choices. A possible corollary which has a form of an existential statement is: 15
Corollary 29 Let (un) and (vn) be two sequences of integers. There exist p < q such that both up uq and vp vq : Proof. Using twice the proposition, we can nd n0 < n1 < : : : such that both extracted
sequences (unk ) and (vnk ) are increasing. We can then take p = n0 and q = n1 . This reasoning can be interpreted in our model. It follows that the following statement is constructively valid (8(un ); (vn))(9p < q )[up uq ^ vp vq ]: In the same way, it would be possible to show in our model the in nite version of Ramsey's theorem [5]. Since the proof-theoretic strength of the theory that we are able interpret is at least the Howard ordinal, we should expect stronger combinatorial principles to be true in the model. But the investigation of this is beyond the scope of this paper.
References
[1] P. Aczel. The type theoretic interpretation of constructive set theory. In: A. Macintyre, L. Pacholski, J. Paris, eds., Logic Colloquium '77, North-Holland 1978. [2] W. Buchholz, S. Feferman, W. Pohlers, W. Sieg, eds. Iterated Inductive De nitions and Subsystems of Analysis: Recent Proof-Theoretical Studies. Springer Lecture Notes in Mathematics, Vol. 897. Springer 1981. [3] T. Coquand. A formal space of ultra lters, manuscript. [4] S. Feferman. Theories of nite type related to mathematicalpractice. In: J. Barwise, ed., Handbook of Mathematical Logic, North-Holland 1977, 913 { 971. [5] R.L. Graham, B.L. Rothschild and J.H. Spencer. Ramsey Theory, 2nd ed. Wiley-Interscience 1990. [6] E.R. Gri or and M. Rathjen. The strength of some Martin-Lof type theories. Archive for Mathematical Logic 33(1994), 347 { 385. [7] P.T. Johnstone. Topos Theory. Academic Press 1977. [8] P.T. Johnstone. Stone Spaces. Cambridge University Press 1982. [9] S. Mac Lane and I. Moerdijk. Sheaves in Geometry and Logic. Springer 1991. [10] P. Martin-Lof. Notes on Constructive Mathematics. Almkvist och Wiksell 1970. [11] E. Palmgren. On universes in type theory. U.U.D.M. Report 1996:19. [12] E. Palmgren. Sheaf-theoretic nonstandard analysis: constructive aspects. U.U.D.M. Report 1996:28. [13] G. Takeuti. Two Applications of Logic to Mathematics. Iwanami Shoten and Princeton University Press 1978. [14] A.S. Troelstra, ed. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer Lecture Notes in Mathematics, Vol. 344. Springer 1973. Thierry Coquand, Department of Computer Science Erik Palmgren, Department of Mathematics Chalmers University of Technology and Gothenburg University S-412 96 Gothenburg, SWEDEN
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