U. Kohlenbach. On the no-counterexample interpretation. The Journal of Symbolic Logic, 64:1491--1511, 1999.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... nite type like appropriate versions of G odel s functional interpretation or modi ed realizability combined with tools like negative translation and or the Friedman Dragalin translation are most useful (in particular compared to techniques which try to avoid any passage through higher types, see [29]) Whereas we have focused on 2) in several publications (see [21] 20] 24] among others) this paper addresses 1) to which S. Feferman has contributed so profoundly. We study mathematical strong, but nevertheless PRA reducible, systems in all nite types emphasizing the need of third order ....

Kohlenbach, U., On the no-counterexample interpretation. 26pp., to appear in: J. Symbolic Logic.

....A(x) where A(x) contains only number quantifiers but maybe function parameters. The solution requires so called bar recursion (of type 0) which was introduced by Spector [118] an which goes beyond Godel s primitive recursive functionals. We will discuss this further in chapters to come (see also [82] for a thorough discussion of the modus ponens problem for the no counterexample interpretation) For the time being we confine ourselves with indicating how the above instance of the modus ponens can be treated if one uses an interpretation which doesn t stop at type level 2: The functional ....

....1. UNWINDING PROOFS: PROOF MINING 17 Suggested further reading 1) On the general program of unwinding proofs (proof mining) 95] 96] 97] 35] 103] 30] 7] 40] 74] 79] 84] 2) On Herbrand s theorem: 23] 40] 70] 113] 95] 102] 3) On the no counterexample interpretation: 40] [82], 90] 91] 119] 120] 111] For other approaches to proof mining not treated in these notes see [26] and [27] Chapter 2 Intuitionistic logic and arithmetic in all finite types In the following we formulate an axiomatic system for intuitionistic first order predicate logic IL. The particular ....

Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic 64, pp. 1491-1511 (1999). BIBLIOGRAPHY 102

....of theorem 4. 21 in [12] we can eliminate F Gamma from the proof of 8u8v tu9w B 0 and extract a uniform bound Phi on 9w which now of course is only in GnR [B 0;1 ] instead of GnR ) and its verification can be carried out in GnA Delta (BR 0;1 ) Pi 0 1 (DC 0 ) By [16] (proposition 4.2) it follows (since deg(fl1) 2) that Phi can be written as a primitive recursive functional Phi such that PA BR 0;1 Phi = fl1 Phi: The final claim follows using again the model M . Since M 0 = S 0 ; M 1 = S 1 and M 2 ae S 2 , the assumption S j= Delta ....

Kohlenbach, U., On the no-counterexample interpretation. To appear in: J. Symbolic Logic.

....not give a procedure for building the terms t 1 , t n . The no counterexample interpretation rather tells which information should be obtained so that using some other technique (as # substitution in the case of Kreisel) we can perform the actual extraction. Moreover, it has been shown in [Koh99] that (contrary to the functional interpretation) the no counterexample interpretation behaves poorly with respect to modus pones. 15 3 Case study: Uniqueness for L 1 approximation The content of this section is based on the article [KO01] which for convenience can be found in the appendix of ....

U. Kohlenbach. On the no-counterexample interpretation. Journal of Symbolic Logic, 64:1491--1511, 1999.

.... finite type like appropriate versions of Godel s functional interpretation or modified realizability combined with tools like negative translation and or the Friedman Dragalin translation are most useful (in particular compared to techniques which try to avoid any passage through higher types, see [28]) Whereas we have focused on 2) in several publications (see [21] 20] 24] among others) this paper addresses 1) to which S. Feferman has contributed so profoundly. We study mathematical strong, but nevertheless PRA redu cible, systems in all finite types, emphasizing the need of third ....

Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic 64, pp. 1491--1511 (1999).

.... finite type like appropriate versions of Godel s functional interpretation or modified realizability combined with tools like negative translation and or the Friedman Dragalin translation are most useful (in particular compared to techniques which try to avoid any passage through higher types, see [28]) Whereas we have focused on 2) in several publications (see [21] 20] 24] among others) this paper addresses 1) to which S. Feferman has contributed so profoundly. We study mathematical strong, but nevertheless PRA reducible, systems in all finite types, emphasizing the need of third ....

Kohlenbach, U., On the no-counterexample interpretation. J. Symbolic Logic 64, pp. 1491-- 1511 (1999). 23

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U. Kohlenbach. On the no-counterexample interpretation. The Journal of Symbolic Logic, 64:1491--1511, 1999.

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U. Kohlenbach. On the no-counter-example interpretation. J. Symbolic Logic 64, pp. 1491-1511 (1999). 16

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