First-Order Reasoning and Primitive Recursive Natural Number Notations

摘要

If the collection of models for the axioms, of elementary number theory (Peano arithmetic) is enlarged to include not just the "natural numbers" or their non-standard infinitistic extensions but also what are here called "primitive recursive notations", questions arise about the reliability of first-order derivations from,. In this enlarged set of "models" some derivations usually accepted as "reliable" may be problematic. This paper criticizes two of these derivations which claim, respectively, to establish the totality of exponentiation and to prove Euclid's theorem about the infinity of primes.