On the No-Counterexample Interpretation

Abstract

In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano
arithmetic. In particular he proved, using a complicated epsilon-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 Phi_A of order type < epsilon_0 which realize the
Herbrand normal form A^H of A.
Subsequently more perspicuous proofs of this fact via functional interpretation (combined
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
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
least not constructively - (gamma) 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 in 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 in L(PA) pointwise in given alpha(< epsilon_0)-recursive functionals satisfying the n.c.i. of A and A -> B.
This yields in particular perspicuous proofs of new uniform versions of the conditions ( gamma), (delta).
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 PA_n by alpha(= 1) is an interpretation in the sense of [17] but not in the sense of [15] since it violates the condition (delta).