G. Kreisel. Mathematical significance of consistency proof. The Journal of Symbolic Logic, 23(2):155--182, 1958.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the n.c.i. of A and A B. Basic Research in Computer Science, Centre of the Danish National Research Foundation. This yields in particular perspicuous proofs of new uniform versions of the conditions (fl) ffi) Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15] 16] In particular we show that the n.c.i. of PAn by ff( n ( recursive functionals (n 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (ffi) 1 Introduction Let ....

....(fl) ffi) Finally we discuss a variant of the concept of an interpretation presented in [17] and show that it is incomparable with the concept studied in [15] 16] In particular we show that the n.c.i. of PAn by ff( n ( recursive functionals (n 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (ffi) 1 Introduction Let 9x A 0 (x; a) be a Sigma 1 formula in the language L(PL) of first order predicate logic PL (a = a 1 ; a k ) are all its free variables) PL 9x A 0 (x; a) then by Herbrand s theorem there are ....

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Kreisel, G., Mathematical significance of consistency proofs. J. Symbolic Logic 23, pp. 155-182 (1958).

....had a very nice feature, namely, from a proof of #x#yA 0 (x, y) where A 0 is a quantifier free formula) in PA (let p denote that proof) by the D interpretation one can obtain in a very modular way a term t and a proof of #xA 0 (x, t(x) in HA # . That was the seeds of Proof Mining. 5 In [Kre58] Kreisel asked for a shift of emphasis in proof theoretic research and announced his program called unwinding of proofs which he described as, There is a different general program which does not seem to suffer the defects of [Hilbert s] consistency program: To determine the constructive ....

G. Kreisel. Mathematical significance of consistency proof. Journal of Symbolic Logic, 23(2):155--182, 1958.

....of affair inevitable Is non confluent reduction intrinsic to classical cut elimination If so, is there a way to manage or control it (just so that we can begin to understand it) Even better, is there a good model of classical cut elimination 2. Extraction of constructive contents Kreisel [20] has shown that any Sigma 0 1 formula that is provable in Peano Arithmetic is also provable in Heyting Arithmetic. This suggests that a constructive interpretation of much of classical logic is plausible. But sev1 eral important questions remain. For example, how should a constructive ....

G. Kreisel. Mathematical significance of consistency proofs. J. Symb. Logic, 23:155--182, 1958.

....its computational features. The restricted system enables us to argue for the expressive power of symmetric and non deterministic calculi like Sym PA . 1 Introduction The possibility of extracting constructive content from classical proofs was established long time ago in seminal papers like [4, 5]. It is only in the last few years, however, that the Computer Science community is trying to effectively and feasibly exploit such a possibility; in particular to establish a Curry Howard correspondence also for classical logic, and hence to widen the scope of the programming with proofs ....

....meaning, i.e. it contains no binder) It is worth noticing that all results of [1] do hold also in case you consider any (consistent) set Delta of atomic rules instead of those for Peano Arithmetic. It is easy to see that Theorem 9 shows an easy, clear way to get what Kreisel obtained in [5], i.e. the extraction of computational contents out of classical proofs. By Theorem 9, from a normalized proof of a disjunction we can get a proof of one of the disjuncts, and from a proof of an existentially quantified formula, we can get a witness of it. More, given a closed proof of a Sigma 0 ....

Kreisel, G. (1958) Mathematical significance of consistency proofs, Journal of Symbolic Logic, 23:155-182.

....for them. There is another interesting use of the above mentioned extended set of reductions: a computational one. Even if it can hardly be thought of as a realistic one for classical logic, which is typically non constructive, the possibility of such a use has been known for a long time. In [5] Kreisel, by means of his no counterexample interpretation for classical proofs, showed that classical and intuitionistic provability coincide if we consider only Sigma 0 1 sentences. Later on Friedman [3] enforced Kreisel s result by providing a translation from classical to intuitionistic ....

G. Kreisel, "Mathematical significance of consistency proofs", Journal of Symbolic Logic, 23:155-182, 1958.

....of finding the computational content of classical proofs. This interest has been motivated by the ambitious purpose of extending to classical logic the well known proofs as programs paradigm of constructive logic. The basis of all the presents investigations are quite old results by Kreisel [9] and Friedman [6] showing that classical and intuitionistic provability coincide for Sigma 0 1 formulas. Now, since from the programming point of view constructivity is needed just for this class of formulas, the problems that the research has to cope with is that of finding techniques to ....

Kreisel G. Mathematical significance of consistency proofs. Journal of Symbolic Logic, 23:155-182, 1958.

....features of classical logic. The problem on which research is beginning to focus now is not the theoretical possibility of having constructive content present in classical proofs, established in old and well known results, but the practical applicability of such results. It was Kreisel in [12] who first pinpointed the presence of constructive content in classical proofs by proving the equality of the sets of Sigma 0 1 sentences provable respectively in intuitionistic and classical logic. Friedman in [7] showed how to get the computational content of a classical proof of a Sigma 0 ....

G. Kreisel, Mathematical significance of consistency proofs, J. Symbolic Logic 23 (1958) 155-182.

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G. Kreisel. Mathematical significance of consistency proof. The Journal of Symbolic Logic, 23(2):155--182, 1958.

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Georg Kreisel. Mathematical significance of consistency proofs. Journal of Symbolic Logic, 23:155--182, 1958.

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G. Kreisel. Mathematical significance of consistency proofs. J. Symbolic Logic 23, pp. 155-182 (1958).

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