  This paper gives a counterexample to the impossibility, by G\"odel'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 G\"odel-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 G\"odel-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.
