Mathematics – Logic
Scientific paper
2001-10-09
Mathematics
Logic
109 pages, preliminary version, pagination added
Scientific paper
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.
Hennix Christer
Yessenin-Volpin Alexander S.
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