A.S. Troelstra, ed., Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, LNM 344, Springer-Verlag, Berlin, 1973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....The formulas x mr # are defined as follows: x mr P P for atomic P D ### # #y(y xy mr #) #y# # #y(xy mr #(y) p 1 x mr #(p 0 x) Note that x mr # implies x D # , for all #. Apart from modified realizability we shall also use Kleene realizability, denoted x r #. See [2] for details. Extend the language of HA by a propositional constant (0 ary relation symbol) U . The theory HA(U) extends HA in the extended language; the only extra axioms are induction axioms for the full extended language. We consider the following translations between HA and HA(U ) 1. The ....

....negative; if # is a quantifier free HA formula, then is almost negative; the almost negative formulas are closed under #, #,#. ECTU is the axiom scheme: #x(A(x) # #yB(x, y) # #z#x(A(x) # zx# B(x, zx) where A(x) must be almost negative. By a trivial adaptation of the methods of [2] we obtain: Next, let us note the following basic properties of the translations ( U and i ) HA(U) ii ) HA iii ) HA(U) iv) HA v) HA U ) vi) HA where in i) ii) v) and vi) # is an HA formula, and in iii) and iv) # is an ....

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A.S. Troelstra, editor. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer, 1973. With contributions by A.S. Troelstra, C.A. Smorynski, J.I. Zucker and W.A. Howard.

....for which S proves #[#] In [9] and [10] he proposed using these methods to identify theorems of classical analysis which may be added as axioms to FIM, to obtain stronger mixed theories S which are consistent relative to their classically correct subtheories and satisfy the rule. Troelstra [17] obtained a nonclassical extension T of FIM with the same properties. The main point of the following pages is to propose an informal axiomatic consistently extending constructive and intuitionistic analysis, containing a virtual image of classical analysis, and closed under Kleene s rule. ....

....formula. In [9] with [11] Kleene showed that if E is any formula in which the scope of every universal function quantifier, and that of every implication, is almost negative, then E is constructively equivalent to the assertion that some function realizes E. Completing this analysis, Troelstra [17] proposed a generalized continuity principle GC 1 relativizing Brouwer s principle of continuous choice to an almost negative hypothesis. Troelstra s Characterization Theorem (for function realizability) says that in the resulting consistent extension T = B GC 1 of FIM, every formula E is ....

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A. S. Troelstra, ed., Metamathematical Investigation of Intuitionistic Arithmetic and Analysis (Springer Verlag, Berlin, Heidelberg, New York, 1973). With contributions by A. S. Troelstra, C. A. Smorynski, J. I. Zucker and W. A. Howard.

.... B) CT 8f :N 9e:N8x:N9y:N: T (e; x; y) U (y) f(x) AC N 8n:N9x:X:R(n; x) 9f :X 8n:N:R(n; f(n) which are true in Mod . On the negative side, we have the failure of Markov s Principle in Mod . In fact, in the presence of IP, CT and AC N, Markov s Principle is inconsistent, see [31]. Since we shall work in the category ModAss of : separated objects of Mod , a few remarks about the internal logic of this category, related to the one of Mod : 1) There is a functor Gamma:Mod Set, left adjoint to r. 2) The regular subobjects in ModAss are precisely those which are : ....

A.S. Troelstra. Metamathematical Investigations in Intuitionistic Arithmetic and Analysis. LNM 344, Springer 1973. 44

....a class of proof like trees which interleave objects from the object language and the metalanguage. The ensuing concept of reflected proof is noteworthy because it is encapsulated in a single recursive type, in contrast to other approaches which result in an infinite tower of types or languages [22, 27] or would result in an infinite tower if extended to provide the same closure [28] Moreover, the issues raised in other uses of reflection are clearly present here, for instance the notions of stance and connection in the procedural reflection of Smith in [26] where he says when you take a step ....

A. S. Troelstra. Metamathematical investigation of intuitionistic mathematics. Lecture Notes in Mathematics, 344, 1973.

....; b; a fl (y; x) for all c all b 2 A c . The symbol fl is now used in various contexts as usual, in particular Gamma fl means that c; a fl whenever c; a fl Gamma. 9.4.8. Theorem. The conditions Gamma fl and Gamma j= are equivalent. More about semantics can be found in [88, 106, 107, 108]. 9.5.1. Exercise. Find constructions for formulas (1) 3) 5) 7) 9) and (11) of Example 9.2.1, and do not find constructions for the other formulas. 9.5.2. Exercise. A first order formula is in prenex normal form iff it begins with a sequence of quantifiers followed by an open formula. ....

A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics. Springer-Verlag, 1973.

....yz 1 : z k ) We only include equality = 0 between numbers as a primitive predicate. E PA is the extension of E PRA obtained by the addition of the schema of full induction and all (impredicative) primitive recursive functionals in the sense of G odel [6] and coincides with Troelstra s [18] system (E HA ) c The weakly extensional versions WE PRA of these systems result if we replace the extensionality axioms (E) by a quanti er free rule of extensionality (due to Spector [17] QF ER: A 0 s = t A 0 r[s] r[t] where A 0 is quanti er free, s ; t ; r[x ] are ....

....zx = 0 zy) We deviate slightly from our notation in [11] The system denoted by E PRA in the present paper results from the corresponding system in [11] if we replace the universal axioms 9) in the de nition of the latter by the schema of quanti er free induction. This terminology is due to [18]. is underivable in WE PA (see [8] The schema of quanti er free choice is given by QF AC : 8x A 0 (x; y) 9Y ( A 0 (x; Y x) QF AC : 2T f QF AC g; where A 0 is a quanti er free formula. In the following we use the formal de nition of the binary ( weak ) K onig s lemma ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

.... also [Kre62] While Kreisel intended to give a consistency proof for the system HA and, accordingly, defined a straightforward extension of Kleene s realizability to this typed system, today s meaning of the term modified realizability derives from Troelstra s collapse of this realizability ([Tro73]) Let me briefly indicate what this is. In Kreisel s notion, one defines for each formula of HA a type ( realizers of have to be found in this type. For example, 9x oe : oe Theta ( and ( Now it is possible to interpret the whole of HA in first ....

....and Streicher s Modified Assemblies turn out to live (as a full subcategory) in Eff Delta Delta rather than Mod. The projectives in Mod are described. Mod is, like many realizability toposes, an exact completion. A generalization of Troelstra s Independence of Premiss principle (see [Tro73]) formulated in the context of Eff Delta Delta , is derived. 1 Definition of Mod and basic properties This section contains some tripos theoretic terminology. In sofar as this remains unexplained, the reader is referred to [HJP80] Convention. From now on we assume the conditions of ....

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A.S. Troelstra, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, LNM 344, Springer 1073 17 Recent Publications in the BRICS Report Series RS-96-3 Jaapvan Oosten. The Modified Realizability Topos. February RS-96-2 Allan Cheng and Mogens Nielsen. Open Maps, Behavioural Equivalences, and Congruences. January

....to E PRA That s why we give a di erent argument for the latter system (which can be adopted also for an alternative proof for the former) In E PRA 1 IA one can show that the continuous functionals ECF form (pointwise) a model of E PRA . Moreover inspection of the proofs of 2.6.6,2.6. 4 in [16] shows that E PRA 1 UB [MUC] ECF , where 1 UB is a restriction of 1 UB (discussed in [12] which allows a direct proof theoretic elimination (see [12] So we have 1 UB [8fA 0 (f ) ECF 1 UB 8fA 0 (f) By the elimination procedure for 1 UB ( 12] thm.4.21) and ....

Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....s maj ae y x s maj x y; xy) Remark 2.2 s maj is a variant of W.A. Howard s relation maj from [6] which is due to [2] For more details see [8] Let A(a) be a formula of G n A (a are all free variables of A) and 9x8yAD (x; y; a) its Godel functional interpretation (see e.g. [24] for details on Godel s functional interpretation) We say that a tuple of closed terms t realizes the monotone functional interpretation of A(a) if ( 9x(t s maj x 8a; y AD (x a; y; a) Monotone functional interpretation which directly extracts a tuple t satisfying ( from a proof of A(a) ....

Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (

.... ; 2 are arbitrary types. WE PA denotes the variant of WE HA with classical logic. In contrast to (E) G odel s functional interpretation trivially satis es QF ER which was introduced in [4] for that very reason. It has been observed in the literature ( 5] 3.5.15 and 1.6. 12) see also [6] for corrections) that WE HA doesn t satisfy the deduction theorem for deductions from open assumptions (whose free variables are treated as parameters and hence are not permitted as proper variables in the quanti er rules as formulated in [5] The argument proceeds as follows: ....

Troelstra, A.S., Metamathematical investigation of intuitionistic arithmetic and analysis. Corrections to the rst edition. ILLC Prepublication Series X-93-04, Universiteit van Amsterdam (

....for deductions from (possibly open) assumptions (with parameters kept xed) was known. Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1 Introduction Let E HA denote the system of extensional intuitionistic arithmetic in all nite types as de ned in [5]. Concerning equality, E HA only contains equality = 0 between numbers as a primitive predicate. For = 0 k : 1 , x 1 = x 2 is de ned as 8y 1 1 ; y k k (x 1 y 1 : y k = 0 x 2 y 1 : y k ) In the context of G odel s functional ( Dialectica ) interpretation, a ....

....arbitrary terms of the system and ; 2 are arbitrary types. WE PA denotes the variant of WE HA with classical logic. In contrast to (E) G odel s functional interpretation trivially satis es QF ER which was introduced in [4] for that very reason. It has been observed in the literature ([5](3.5.15 and 1.6.12) see also [6] for corrections) that WE HA doesn t satisfy the deduction theorem for deductions from open assumptions (whose free variables are treated as parameters and hence are not permitted as proper variables in the quanti er rules as formulated in [5] The ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....n.c.i. of A and A B. A third way to prove the no counterexample interpretation of PA (by functionals which are ff( 0 ) recursive) is via Godel s functional interpretation (combined with negative translation) of PA in the calculus T of primitive recursive functionals of finite type (see e.g. [31]) This (combination of negative translation and) functional interpretation is a local A formalization of the method of substitution was given by [29] and used in [20] thm.12) One should also mention here Godel s discussion of Gentzen s 1936 consistency proof in his amazing Vortrag bei ....

.... B but only on the quantifier prefix of their prenex normal forms) This is In connection with [3] one should mention that some of the result obtained in this paper by minimal realizability can in fact be derived (sometimes in much stronger form) using only well known facts from the literature ([31], 24] see [14] Tn denotes the fragment of Godel s T (see [7] with R ae for deg(ae) n only. c PA is the subsystem of PA based on T0 instead of T and with quantifier free induction only, see [4] and section 2 below. true for A 2 Pi 3 and B 2 Pi 1 (but already in this case T ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (

....= 0) ii) A is closed under computation in the sense of Kleene s schemata S1 S9. iii) there exists a 2 A 2 such that has no associate in A 1 . By (ii) A is a model of the restriction of E PA to the fragment with pure types only. Modulo the well known reduction to pure types (see [37](1.8.5 1.8.8) E PA therefore has a model in which there exists a functional which has no associate and therefore by the previous proposition no r.m. code f . Nevertheless, all functionals of type 2 are continuous: one could use here an argument due to [14] to show that the ....

Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

.... f # C[0, 1] in the sense of [10] 11] Before going into the details of the analysis we need to recall some general logical background from [15] 5 First we introduce a little amount of logical terminology: Let A # be a (sub )system of arithmetic in all finite types (like E PA # from [26] or Feferman s fragment E PRA # with quantifier free induction and primitive recursion on the type 0 only [8] Let A # # denote the extension of A # by the schema QF AC : #f 1 #x 0 A qf (f, x) # #F 2 #f 1 A qf (f, F (f) of quantifier free choice from functions to numbers ....

A.S. Troelstra. Metamathematical investigation of intuitionistic Arithmetic and Analysis, Springer Lecture Notes in Mathematics, volume 344, pages 1--323. Springer Verlag, Berlin, 1973.

....the well known Ackermann function. Show that # is definable in WE HA # by a closed term t 0(0) 0) i.e. WE HA # proves the equations (#) for t) Suggested further reading: See [121] for further information on the BHK interpretation. For more information on (W)E HA # and its variants see [122]. Chapter 3 Modified realizability Definition 3.1 (modified realizability) For each formula A of E HA # we define a formula x mr A (in words: x modified realizes A ) of E HA # whose free variables are contained in that of A and x, where x is a possibly empty tuple of variables which ....

....in remark 3. 3) every # free formula A is already in E HA # provably equivalent to A and hence to a negated formula (here one uses that the prime formulas of E HA # are decidable) This even holds for the more general class of so called Harrop formulas instead of # free formulas (see [122](1.10.5 1.10.8) so that Harrop formulas are covered as well by our set of axioms #. Definition 3.12 ( 122] The subset # 1 of formulas # L(E HA # ) is defined inductively by 1) Prime formulas are in # 1 . 2 2) A, B # # 1 # A # B, A # B, #x A(x) #x A(x) # # 1 . 3) If A is ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....has been an open problem whether WMP is derivable in Bishop style mathematics. The problem is, of course, not completely precise as no particular formal system has been identified with Bishop style mathematics. However, it is commonly agreed that Heyting arithmetic in all finite types HA # (see [17]) plus the axiom of choice AC in all types AC #,# : #x # #y # A(x, y) # #Y ### #x # A(x, Y (x) is a framework which is quite capable of formalizing existing constructive (in the sense of Bishop) mathematics (see also [1] 6] In this note we show that WMP is underivable even in E HA ....

.... which is quite capable of formalizing existing constructive (in the sense of Bishop) mathematics (see also [1] 6] In this note we show that WMP is underivable even in E HA # AC, where E HA # is Heyting arithmetic in all finite types with the full axiom of extensionality (see again [17] for a precise definition) Our proof even establishes that this underivability 2 remains true if the (highly non constructive) schema of full comprehension in all types for arbitrary negated formulas CA : ## ##0 #x # (#(x) 0 0 # A(x) is added to E HA # AC which e.g. allows to ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....ae y x y s maj x y; xy) 6 Remark 2.2 s maj is a variant of W.A. Howard s relation maj from [6] which is due to [2] For more details see [8] Let A(a) be a formula of GnA (a are all free variables of A) and 9x8yAD (x; y; a) its Godel functional interpretation (see e.g. [25] for details on Godel s functional interpretation) We say that a tuple of closed terms t realizes the monotone functional interpretation of A(a) if 7 ( 9x Gamma t s maj x 8a; y AD (x a; y; a) Delta (Monotone functional interpretation which directly extracts a tuple t satisfying ( from a ....

Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973). 28

....(resp. a finitely iterated exponential function) in u M and k. The methods by which our extraction of bounds is achieved are monotone versions of the so called modified realizability interpretations mr and mrt. Modified realizability was introduced in [13] and is studied in great detail in [14] and [16] to which we refer) 1 In [14] 16] these interpretations are developed for theories like E HA (and immediately apply also to E PA i and E PRA i ) Furthermore both interpretations apply to our theories E Gn A i : The interpretation of the logical part can be carried ....

....in u M and k. The methods by which our extraction of bounds is achieved are monotone versions of the so called modified realizability interpretations mr and mrt. Modified realizability was introduced in [13] and is studied in great detail in [14] and [16] to which we refer) 1 In [14], 16] these interpretations are developed for theories like E HA (and immediately apply also to E PA i and E PRA i ) Furthermore both interpretations apply to our theories E Gn A i : The interpretation of the logical part can be carried out using only Pi ae; Sigma ffi;ae; ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

.... The predicate T (x, y, z) is decidable (even primitive recursive) and the predicate T # (x) #yT (x, x, y) is undecidable (but recursively enumerable) The material of the following section on Proof Mining is substantially based on [AF98] BS95] BSBar] Bus95] Koh98a] Koh93a] [Tro73] and [Tv88] 4 2 Proof Mining The general purpose of Proof Mining is to extract from a given proof of a formula A in a system A some constructive content. By constructive content we normally mean a realizing term for the existential quantifiers of A. For instance, if A : #x#yA 0 (x, y) ....

A.S. Troelstra. Metamathematical investigation of intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, volume 344, pages 1--323. Springer Verlag, 1973.

....we can take F (u) m2N fL 2 S 0 j u = f m xg and we have 1 = F (t c n x f) which implies t c n x f = f m x for some m. 4 An Application We work now in SAS 0 : second order arithmetic with arithmetical comprehension. It is known that this system is conservative over Peano Arithmetic [Troe]. It is possible to represent D and the poset S 0 in SAS 0 . The argument above however cannot be formalised as it is in SAS 0 because of the lemma 4 which requires the de nition of semantics of formulae. We consider a xed derivation of a typing judgement of the form t : N N . In this ....

....nite set of quanti ed types T 1 ; Tn and we consider the set SF of subformulae of C T1 (t 1 ) C Tn (t n ) We let then S 1 be the subposet of S 0 which consists only of nite sets of such subformulae. Given any poset de ned in SAS 0 we can de ne [ A] for A 2 SF in SAS 0 , see [Troe], p. 37. Lemma 7. If M A with A 2 SF; M SF then [ M ] A] and this is provable in SAS 0 : We consider then the model H = Down(S 1 ) and 1 = S 1 ; X Y = X Y; X Y = fL 2 S 1 j M 2 X L; M 2 Y g for all X;Y D. Corollary 2. If A is a rst order formula in SF with at most X ....

A. Troelstra. Metamathematical Investigations of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics 344, 1973.

....may be chosen arbitrarily. As for quanti ers, the statement r realises 9xA can be de ned simple to mean 9x (r realises A) similarly for 8) i.e. quanti ers are ignored , i.p. the x is not part of the realiser r. Note that this is closely related to realisability for second order arithmetic [60]. The standard treatment of the quanti ers can be recovered as a special case by reading e.g. 9x N A as 9x(N(x) A) and interpreting r realises N(x) as r = x. However, the usefulness of these modi cations lies in the interpretation of e.g. r realises R(x) R for the reals) which ....

A.S. Troelstra. Metamathematical Investigations of Intuitionistic Arithmetic and Analysis. LNM 344, 1973.

....defined notion: A :# A # #. Finally L(HL # ) contains logical combinators # #,# and # #,#,# of type ### and ##(##) ###) for all #, #, # # T. HL # has the usual axioms and rules of intuitionistic predicate logic (for all sorts of variables) plus the equality axioms for = 0 (e.g. see [34] ) Equations s = # t between terms of higher type # = 0# k . # 1 are abbreviations for the formulas #x #1 1 , x #k k (sx 1 . x k = 0 tx 1 . x k ) # #,# , # #,#,# are characterized by the corresponding axioms of typed combinatory logic: # #,# x # y # = # x and # #,#,# ....

....A # A is added to HL # . The enrichment of HL # (resp. PL # ) obtained by adding the extensionality axiom (E # ) #x # , y # , z ## (x = # y # zx = # zy) for every type # is denoted by E HL # (resp. E PL # ) Remark 2.1. 1 Using # #,# and # #,#,# one defines (e.g. as in [34] ) # terms #x # .t # [x] for each term t # [x # ] such that HL # # #x # .t # [x] s # = # t[s] In particular one can define a combinator # # #,# = #x # , y # .y such that # # #,# x # y # = # y (e.g. take # # : #(###) for #, # of suitable types) Notational ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....takes an exact completion of the result. This also has the e ect of improving the categorical properties. Formally the main result of the paper is that the result is a topos just (modulo some technical conditions) when the original category has a universal object. Early work on realizability (e.g.[12, 22], or see [23] is characterised by its largely syntactic nature. The core de nition is when a sentence of some formal logic is realised, and the main interest is in when certain deductive principles (such as Markov s rule) are validated. Martin Hyland s invention y The authors wish to ....

A.S. Troelstra, editor. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer, 1973. With contributions by A.S. Troelstra, C.A. Smorynski, J.I. Zucker and W.A. Howard.

....method in general and then give a realizer for the (simpler) classical axiom of countable choice. 3.1. Witnesses from classical proofs. The method we use to extract witnesses from classical proofs is a combination of G odel s negative translation (translation P o in [15] page 42, see also [24]) the Dragalin Friedman Leivant trick, also called A translation [23] and Kreisel s (formalized) modi ed realizability [22] The method works in general for proofs in classical arithmetic in nite types, PA (for simplicity without extensionality, for the problem of eliminating extensionality ....

A.S. Troelstra, Metamathematical investigation of intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, pp. 1-323, Springer Verlag, 1973, pp. 1{

....sense of [11] 12] 1. 1 Logical background Before going into the details of the analysis we need to recall some general logical background from [16] 5 First we introduce a little amount of logical terminology: Let A # be a (sub )system of arithmetic in all finite types (like E PA # from [27] or Feferman s fragment E PRA # with quantifier free induction and primitive recursion on the type 0 only [8] Let A # # denote the extension of A # by the schema QF AC : #f 1 #x 0 A qf (f, x) # #F 2 #f 1 A qf (f, F (f) of quantifier free choice from functions to numbers ....

A.S. Troelstra. Metamathematical investigation of intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, volume 344, pages 1--323. Springer Verlag, 1973.

....we define a notion of extensionality for higher order logic programs and show that for extensional programs the semantics can be considerably simplified. This so called extensional collapse originates from the model theory of finite type arithmetic and is described and attributed to Zucker in [12]. Definition 6.1 Let P be a typable higher order logic program, with P and [ P as in Definition 4.1. We define relations # # P on # P , expressing extensional equality of type #. DD: We put # # P to be = equality on # P , for every # # DD. 6. The extensional collapse 9 TT ....

A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Number 344 in Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1973.

....extracting function, see [68] In this form CT states that if P is a decidable predicate (i.e. the excluded middle 5 Is it decidable whether a given Diophantine equation has a solution in the integers 188 HENK BARENDREGT holds for P) then P has a recursive characteristic function. See [112] for formal consequences, models, counter models and an extension of CT. 3. Computing. Lambda calculi are prototype programming languages. As is the case with imperative programming languages, where several examples are untyped (machine code, assembler, Basic) and several are typed (Algol 68, ....

A. S. Troelstra (editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin and New York, THE IMPACT OF THE LAMBDA CALCULUS IN LOGIC AND COMPUTER SCIENCE 215 1973.

.... ]R(n; f(n) for some f : N N by AC in F ; E 8n:N: R(n; f(n) by stability; E 8n:N: R(n; f(n) by ml 1: 2) As in (Birkedal, 2000) we show that if the topos of discrete objects satis es the arithmetic form of Church s Thesis (in the sense of, for instance, Troelstra and van Dalen, 1988; Troelstra, 1973)) then a ] ed version is satis ed by E . Observe that E has a natural numbers object if and only if D j E does, because both S. Awodey and L. Birkedal and D.S. Scott 12 : D j E E and : E D j E are inverse images of geometric morphisms, which preserve the natural numbers object (Johnstone, ....

Troelstra, A., editor (1973). Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer. With contributions by A.S. Troelstra, C.A. Smorynski, J.I. Zucker and W.A. Howard.

....w i v w[ x 0 ; u i ) Therefore t i (x; u; w) t i ( x; u) w) #t. Clearly t i is total. 2 This theorem immediately implies that for nite types s, the set I tot (s) of total elements of type s is dense in I(s) This has been proved rst by Kleene [7] and Kreisel [8] see also [23], 2.6.19 and [6] In [3] the notion of co density of a subset of a domain (called totality there) was introduced and a corresponding theorem with and instead of and was proved. 3.4 Density for universes Now we are going to show that the wellfounded universe (S wf ; I tot ) is dense ....

....(I tot (s) I tot f) is dense as well. Hence we may apply the proposition above with (D ; F ) I tot (s) I tot f ) 2 This theorem shows that the hierarchy (S wf ; I tot ) satis es a quanti er free axiom of choice. For nite type this has been established already by Kreisel [8] see also [23], 2.6.20) 4.2 Extensionality In our next application we show that in the total universe (S wf ; I tot ) extensional equality coincides with consistency in the domain ordering, and as a consequence that all well founded types and their total objects are extensional, i.e. respect extensional ....

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A. S. Troelstra. Metamathematical Investigations of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, 344, Springer, 1973.

.... of a statement is given by the set of its realizers (that is, its possible or candidate proofs if any, as this set may be empty) This constructive understanding of logical systems is also related the so called BHK explanation of the intuistionistic meaning of the logical connectives (see [Troe73], TroVan73] for both approaches) In either semantics, one constructively gives meaning to a statement, to a defined mathematical concept, by discussing its provability, as truth is effective provability (or provability by proof terms) In this perspective, section 4 below presents a rigorous ....

A. Troelstra. Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics 344, Springer Verlag, 1973.

....relative to the base system RCA # 0 instead of RCA 2 0 : Proposition 3.1. RCA # 0 is a conservative extension of RCA 2 0 . HIGHER ORDER REVERSE MATHEMATICS 5 Proof: Locally, one can show in RCA 2 0 that the type structure ECF of all extensional hereditarily continuous functionals (see [17] for the technical definition) forms a model of RCA # 0 , i.e. 1) RCA # 0 # A # RCA 2 0 # [A] ECF . Together with the fact that (2) RCA 2 0 # #f 1 (#(f) 0 [#] ECF (f) for all ordinary primitive recursive functionals # 2 of type 2 (i.e. the functionals definable in RCA 2 0 ) ....

....(2) RCA 2 0 # #f 1 (#(f) 0 [#] ECF (f) for all ordinary primitive recursive functionals # 2 of type 2 (i.e. the functionals definable in RCA 2 0 ) this yields the conservation result. 1) is proved similarly to (and in fact easier than) the corresponding result for E HA # QF AC from [17](2.6.20) In particular, no induction beyond # 0 1 IA is needed. # As a corollary of proposition 3.1 we get that the finite type extensions RCA # 0 WKL and RCA # 0 # 0 # CA etc. of the second order systems used in reverse mathematics are conservative over their second order part. ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....A is closed under computation in the sense of Kleene s schemata S1 S9. iii) there exists a # # A 2 such that # has no associate in A 1 . By (ii) A is a model of the restriction of E PA # QF AC 1,0 to the fragment with pure types only. Modulo the well known reduction to pure types (see [38](1.8.5 1.8.8) E PA # QF AC 1,0 therefore has a model in which there exists a functional # 2 which has no associate and therefore by the previous proposition no r.m. code f . Nevertheless, all functionals # 2 of type 2 are continuous: one could use here an argument due to [14] to ....

....continuous functionals # 1(1) have an r.m. code is # 1 # conservative over E PA # QF AC 0,0 . Proof: Formalizing the fact that the extensional continuous functionals ECF form a model of the first theory and the proof for the faithfulness of this model for the analytical fragment (see [38](2.6.5 2.6.12) a proof of A # # 1 # in the first theory translates into a proof of [A] ECF (and hence of A) in E PA # QFAC 0,0 . # Corollary 4.9. E PA # QF AC all continuous functionals # 1(1) have an r.m. code proves neither # 0 # CA nor WKL. Proof: The corollary follows ....

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

....D. Note that the ( direction of both equivalences hold in intuitionistic logic. These equivalences are natural ones to choose given that we expect that goals which are operationally equivalent are logically equivalent. The ) directions of these rules are known as the Independence of Premise axioms [13], and although there are some results regarding the addition of these and similar axioms to Heyting arithmetic [13] not much seems to be known about the logic obtained by adding these rules to intuitionistic logic. We may think of such a logic as a logic of present choice , in that the choice ....

....to choose given that we expect that goals which are operationally equivalent are logically equivalent. The ) directions of these rules are known as the Independence of Premise axioms [13] and although there are some results regarding the addition of these and similar axioms to Heyting arithmetic [13], not much seems to be known about the logic obtained by adding these rules to intuitionistic logic. We may think of such a logic as a logic of present choice , in that the choice of witness for the existentially quantified variable cannot be postponed; if we can ever choose such a witness, then ....

A.S. Troelstra, Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics 344, SpringerVerlag, Berlin, 1973.

....is easily expressed using universally quantified conditional equations. We use a technique of Lambek[18] involving Mal cev operators to equationally express uniqueness of iteration (more generally, higher order primitive recursion) in a simply typed lambda calculus, essentially Godel s T [29, 13]. We prove the following facts about typed lambda calculus with uniqueness for primitive recursors: i) It is undecidable, ii) Church Rosser fails, although ground Church Rosser holds, iii) strong normalization (termination) is still valid. This entails the undecidability of the coherence ....

.... primitive recursor, Rahxy, where a : A ) B;h : A ) N ) B ) B, satisfying: Rahx0 = ax Rahx(Sy) hxy(Rahxy) Again we may ask: how do we know such primitive recursive definitions uniquely specify the intended function Such uniqueness questions make sense in many contexts: arithmetic theories [29], first order term rewriting theories [9] primitive recursive arithmetics [14] simply and higher order typed lambda calculi and related functional languages[13, 20, 29] categorical programming languages [5, 15] and more generally wherever we define a procedure iteratively on an inductive data ....

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A. S. Troelstra. Metamathematical investigations of intuitionistic arithmetic and analysis, Springer LNM 344, 1973. Theory and Applications of Categories, Vol. 6, No. 4 64

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A. S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Chapters 1--4. Lecture Notes in Mathematics 344. Springer-Verlag, Berlin, Heidelberg, New York, 1973.

....to the modality # just as the constructors in the ordinary typed lambda calculus correspond to in propositional formulas. They gave a natural deduction system for PLL and prove a strong normalization theorem by using the method in Prawitz [Pra97] see also Tait [Tai67] and Troelstra [Tro73]) Gol81] argued for an application of the logic in Grothendieck s topology. He extracted the principle (#) A is locally true at # i# A is true at all points close to # For instance, two functions f and g are said to be equivalent, or to have the same germ, at a point # in the intersection of ....

A. S. Troelstra, Metamathematical investigations of intuitionistic arithmetic and analysis, Lecture notes in Mathematics 344, Springer-Verlag, 1973.

....and say something concrete about what is in this paper. According to me, there are three landmark publications in Realizability. These are: 1) Kleene s original 1945 paper, On the Interpretation of Intuitionistic Number Theory ( 51] 2) Troelstra s Metamathematical Investigations from 1973 ([93]) 3) Hyland s The Effective Topos from 1981 ( 40] Of these three, both 1) and 3) initiated a whole new strand of research. I have therefore decided that the material I wished to present, naturally divides into two periods, viz. 1940 1980 and 1980 2000. This is not to say that suddenly there ....

....just conceivable that he tried: a realizer for A B is a partial recursive function which sends proofs of A to proofs of B. Kleene s realizability was, at least conceptually, a major advance. Its achievement is not so much a philosophical explanation of the intuitionistic connectives. Troelstra ([93], p.188) says: it cannot be said to make the intended meaning of the logical operators more precise. As a philosophical reduction of the interpretation of the logical operators it is also only moderately successful; e.g. negative formulae are essentially interpreted by themselves. In fact, ....

[Article contains additional citation context not shown here]

A.S. Troelstra, editor. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Springer, 1973. With contributions by A.S. Troelstra, C.A. Smory'nski, J.I. Zucker and W.A. Howard.

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A.S. Troelstra, ed., Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, LNM 344, Springer-Verlag, Berlin, 1973.

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A.S. Troelstra, 1973. Metamathematical investigations of intuitionistic arithmetic and analysis. Springer-Verlag Lecture Notes in Mathematics 344.

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A.S. Troelstra (ed.) Metamathematical investigations of intuitionistic arithmetic and analysis. second, corrected edition, 1993, first edition 1973 ILLC Prepublication series X93 -05, University of Amsterdam pp. 147 e.v.

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A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis,vol- ume 344 of Lecture Notes in Mathematics. Springer Verlag, 1973.

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A. Troelstra, editor. Metamathematical investigations of intuitionistic arithmetic and analysis, volume 344 of Springer Lecture Notes in Mathematics. Springer, Berlin, Heidelberg, New York, 1973.

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A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics. Springer Verlag, 1973.

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A. S. Troelstra. Metamathematical investigation of intuitionistic Arithmetic and Analysis, volume 344 of Lecture Notes in Mathematics. Springer Verlag, 1973.

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A. S. Troelstra, Metamathematical investigation of intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics, vol. 344, Springer, Berlin, 1973.

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

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A. Troelstra. Metamathematical Investigation of Intuitionistic Mathematics, Lecture Notes in Mathematics, Vol. 344. Springer-Verlag, 1973.

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Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic arithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973).

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A.S. Troelstra, Metamathematical Investigations of Intuitionistic Arithmetic and Analysis, Lecture Notes in Mathematics 344, Springer-Verlag, Berlin, 1973. 32

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A.S. Troelstra. Metamathematical Investigation of Intuitionistic Arithmetic and Analysis. Lecture Notes in Mathematics, No. 344, Springer-Verlag, 1973. 18

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