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.

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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.

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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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