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- W2259673208 abstract "In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano<br />arithmetic. In particular he proved, using a complicated epsilon-substitution method (due to<br />W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic (PA) one can find ordinal recursive functionals Phi_A of order type < epsilon_0 which realize the<br />Herbrand normal form A^H of A.<br /> Subsequently more perspicuous proofs of this fact via functional interpretation (combined<br />with normalization) and cut-elimination were found. These proofs however do not carry out the n.c.i. as a local proof interpretation and don't respect the modus<br />ponens on the level of the n.c.i. of formulas A and A -> B. Closely related to this phenomenon is the fact that both proofs do not establish the condition (delta) and - at<br />least not constructively - (gamma) which are part of the definition of an `interpretation of a<br />formal system' as formulated in [15].<br />In this paper we determine the complexity of the n.c.i. of the modus ponens rule for<br />(i) PA-provable sentences,<br />(ii) for arbitrary sentences A;B in L(PA) uniformly in functionals satisfying the n.c.i. of (prenex normal forms of) A and A -> B; and<br />(iii) for arbitrary A;B in L(PA) pointwise in given alpha(< epsilon_0)-recursive functionals satisfying the n.c.i. of A and A -> B. <br /> This yields in particular perspicuous proofs of new uniform versions of the conditions ( gamma), (delta). <br /> Finally we discuss a variant of the concept of an interpretation presented in [17] and<br />show that it is incomparable with the concept studied in [15],[16]. In particular we show<br />that the n.c.i. of PA_n by alpha(<omega_n(omega))-recursive functionals (n >= 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (delta)." @default.
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- W2259673208 date "1997-06-12" @default.
- W2259673208 modified "2023-10-01" @default.
- W2259673208 title "On the No-Counterexample Interpretation" @default.
- W2259673208 doi "https://doi.org/10.7146/brics.v4i42.18968" @default.
- W2259673208 hasPublicationYear "1997" @default.
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