A.S. Troelstra "Realizability". In S. Buss, ed., Handbook of Proof Theory, Elsevier, pp. 407-474, 1998. Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
Explicit Provability: The Intended Semantics for Intuitionistic.. - Artemov (1998) (Correct)
....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 between intuitionistic and classical understanding of realizability semantics for intuitionistic theories. Intuitionistic realizability enjoys some nice ....
....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 example, 24] 20] 72] The inductive definition of the Curry Howard isomorphism goes along the lines of BHK clauses, where ....
A.S. Troelstra "Realizability". In S. Buss, ed., Handbook of Proof Theory, Elsevier, pp. 407-474, 1998.
Operations on Proofs That Can Be Specified By Means of Modal Logic - Artemov (Correct)
....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 derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [23] turned out to be based on an inconsistent ....
....Cornell University, email:artemov math.cornell.edu and Moscow University, Russia. not derivable in Int. A formalization of the BHK semantics suggested by Kreisel in [23] turned out to be based on an inconsistent theory (cf. 48] 37] For more discussion on realizability semntics for Int see [45]. In 1933 Godel ( 15] defined Int on the basis of the notion of proof in a classical mathematical system, reminiscent to the one from the classical BHK semantics. Namely, Godel introduced the logic of provability (coinciding with the modal logic S4) and constructed an embedding of Int into S4. In ....
A.S. Troelstra "Realizability". In S. Buss, ed., Handbook of Proof Theory, Elsevier, pp. 407-474, 1998.
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