G. Kreisel. On weak completeness of intuitionistic predicate logic. The Journal of Symbolic Logic, 27:139--158, 1962.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Some important subcategories of Mod are described, and a general logical principle is derived, which holds in the larger topos and implies the well known Independence of Premiss principle. Introduction. The notion of modified realizability originates with Kreisel s [Kre57] see 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] ....

G. Kreisel, On weak completeness of intuitionistic predicate logic, JSL 27 (1962), Number 2 (June), pp. 139-158

....version of realizability, the so called recursive realizability, was introduced by Kleene CHAPTER 2. INTUITIONISTIC ARITHMETIC 21 in [65] In these lectures we will focus on a typed variant of Kleene s type free interpretation which is called modified realizability and is due to Kreisel [92] [93]. Recently S. Artemov has developed a so called logic of proofs where proof in the BHK clauses is interpreted as t is a proof (polynomial) for A referring to some standard proof (not: provability) predicate e.g. for PA. Using this interpretation he proves a completeness result for ....

Kreisel, G., On weak completeness of intuitionistic predicate logic. J. Symbolic Logic 27, pp.139--158 (1962).

....i.e. Phiuk is bounded by a polynomial (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 ....

Kreisel, G., On weak completeness of intuitionistic predicate logic. J. Symbolic Logic 27, pp.139-- 158 (1962).

....sentence A a notion the number n realizes A . That turned out to be a systematic method of making the constructive content of arithmetical sentences explicit. We present here a version of Kleene s realizability for E HA # , called modified realizability, which was first formulated by Kreisel [Kre62]. In the following x stands for a tuple (possibly empty) which is built from variables in the language of E HA # and tupling operations. Definition 2.4 (Modified realizability) With each formula A # L(E HA # ) associate a formula x mr A (read: x modified realizes A) of E HA # by ....

G. Kreisel. On weak completeness of intuitionistic predicate logic. Journal of Symbolic Logic, 27:139--158, 1962.

....est satisfaite. On notera, a ce sujet, que des preuves intuitionnistes du theoreme de completude de la logique intuitionniste, utilisant des variantes de la notion de modele de Kripke, ont ete donnees par H. Friedman [7] et W. Veldman [8] On remarquera egalement qu un argument de G. Kreisel [2] montre que le theoreme de completude habituel de la logique intuitionniste n a pas de preuve intuitionniste. La seconde etape est abordee dans la section III, ou l on se contente d enoncer les resultats ; les preuves seront developpees dans un article ulterieur. Cette etape consiste a analyser ....

G. Kreisel. On weak completeness of intuitionistic predicate logic. Journal of Symbolic Logic 27 (1962) p. 139-158.

....for intuitionistic theories, first of all realizability. The basic notions of realizability were defined along the lines of BHK clauses with different constructive objects instead of proofs: computable functions and their codes (e.g. in [32] 33] computable operations of higher types (e.g. in [38]) partial recursive operations (e.g. in [21] 22] etc. For the references one may consult recent surveys on realizability and constructive semantics [8] 71] Note that the standard realizability semantics for Int is not adequate. First of all, following Kleene ( 32] one should distinguish ....

....of intuitionistic arithmetic. Such a result relates Int with a formal theory based on the same Int and thus is not intended to give an independent semantics for the latter. On the other hand, classical realizabilities (Kleene realizability [32] function realizability [33] modified realizability [38], Medvedev s calculus of finite problems [50] and its variants) give conditions necessary but not sufficient for Int(cf. 18] 71] 74] 75] It turned out that the natural deduction proofs for Int can be transliterated by the CurryHoward isomorphism into the language of typed terms (see, for ....

G. Kreisel, "On weak completeness of intuitionistic predicate logic", Journal of Symbolic Logic, v. 27, pp. 139-158, 1962.

....of classical proofs, thus providing an independent definition of intuitionistic logic within the classical mathematics. We call this sort of interpretation of Int classical BHK semantics. Classical realizabilities: Kleene realizability [19] function realizability [20] modified realizability [24], Medvedev s calculus of finite problems [31] and its variants, give conditions necessary but not sufficient for Int(cf. 12] 45] 46] 47] Each of them realizes some formulas Department of Mathematics, Cornell University, email:artemov math.cornell.edu and Moscow University, Russia. not ....

G. Kreisel, "On weak completeness of intuitionistic predicate logic", Journal of Symbolic Logic, v. 27, pp. 139-158, 1962.

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G. Kreisel. On weak completeness of intuitionistic predicate logic. The Journal of Symbolic Logic, 27:139--158, 1962.

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