Beware of the Gödel- Wette paradox

Abstract

This paper gives a counterexample to the impossibility, by Gödel's second incompleteness theorem, of proving a formula expressing the consistency of arithmetic in a fragment of arithmetic on the assumption that the latter is consistent. This counterexample gives rise to a new type of metamathematical paradox, to be called the Gödel-Wette paradox, which E. Wette claims to have established since some time ago (see [Wette, 1971], [Wette, 1974]). Nevertheless, our work is independent of Wette's since we have failed to understand the details of his work although we recognize the possibility of the correctness of the latter. Furthermore, the Gödel-Wette paradox is not the only foundational anomaly which the framework of our approach has uncovered but new questions concerning the decision problem, completeness problem, truth definitions and the status of Richard's paradox in arithmetic and set theory (including type theory) have arisen as well. This work will, eventually be unified into a single monograph.