On the No-Counterexample Interpretation

In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated e-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals ΦA of order type < e0 which realize the Herbrand normal form A of A. Subsequently more perspicuous proofs of this fact via functional interpretation (combined with normalization) and cut-elimination where found. These proofs however do not carry out the n.c.i. as a local proof interpretation and don’t respect the modus 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 (δ) and – at least not constructively – (γ) which are part of the definition of an ‘interpretation of a formal system’ as formulated in [15]. In this paper we determine the complexity of the n.c.i. of the modus ponens rule for (i) PA-provable sentences, (ii) for arbitrary sentences A,B ∈ L(PA) uniformly in functionals satisfying the n.c.i. of (prenex normal forms of) A and A→ B, and (iii) for arbitrary A,B ∈ L(PA) pointwise in given α(< e0)-recursive functionals satisfying the n.c.i. of A and A→ B. ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.