AbstractIn [15], [16] G. Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated ε-substitution method (due to W. Ackermann), that for every theoremA(Aprenex) of first-order Peano arithmeticPAone can find ordinal recursive functionalsof order type < ε0which realize the Herbrand normal formAHofA.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 no-counterexample interpretation as alocalproof interpretation and don't respect the modus ponens on the level of the nocounterexample interpretation of formulasAandA → 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].