S. C. Kleene, Classical extensions of intuitionistic mathematics, in: ICM 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... to tying down his relative consistency proof, this work established that B, FIM, and other suitable extensions of M obey a recursive uniformization rule: If ####R(#,#) is a closed theorem of the extended theory S, then there is a recursive total functional #[#] 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 ....

....hierarchy does not collapse. The only nonclassical axiom of due to Troelstra, depends on a syntactically defined class of relations isolated by Kleene. A formula is almost negative if it contains no disjunction, and no existential quantifier except immediately before a prime 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 ....

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S. C. Kleene, Classical extensions of intuitionistic mathematics, in: ICM 1964.

....gave rise to intensive studies of constructive semantics 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 ....

....in a certain extension HA 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 ....

S. Kleene. "Classical extensions of intuitionistic mathematics", In Y. Bar-Hillel, ed. Logic, Methodology and Philosophy of Science 2, North Holland, pp. 31-44, 1965

....to interpret Int on the basis 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 ....

S. Kleene. "Classical extensions of intuitionistic mathematics", In Y. Bar-Hillel, ed. Logic, Methodology and Philosophy of Science 2, North Holland, pp. 31-44, 1965

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