This paper proposes to replace \sys{PA}, Peano Arithmetic, by a theory \sys{APA} defined in terms of (i) a set of axioms that is classically equivalent to the Peano axioms and (ii) a defeasible logic that minimizes inconsistency, viz.\ an inconsistency-adaptive logic. If \sys{PA} is consistent, its set of theorems coincides with the set of \sys{APA}-theorems. If \sys{PA} is inconsistent, \sys{APA} is non-trivial and has the following remarkable property: there is a unique non-standard number that is its own successor and every `desirable' \sys{PA}-theorem is retained if restricted to the other numbers. The restriction can be expressed in the language of arithmetic. And there is much more.

%B Modestly Radical or Radically Modest. Festschrift for Jean Paul Van Bendegem on the Occasion of His 60th Birthday %I College Publications %P 11–59 %G eng