Some Transfinite Generalisations of Gödel's Incompleteness Theorem

On the Results of a 14-Year Effort to Generalize Gödel’s Second Incompleteness Theorem and Explore Its Partial Exceptions

The Second Incompleteness Theorem states that axiom systems of sufficient strength are unable to verify their own consistency. Part of the reason that Gödel's theorem has fascinated logicians is that it almost defies common sense. This is because when human beings cogitate, they implicitly presume that thinking is a useful process. However, this tacit assumption would initially appear to presum...

IPM-Isfahan workshop on Various Aspects of Modality

Paradoxes are amazing puzzles in philosophy (and mathematics), and they can be interesting when turned into proofs and theorems (in mathematics and logic). For example, Russell's paradox, which collapsed Frege's foundations of mathematics, is now a classical theorem in set theory, and implies that no set of all sets can exist. Or, as another example, the Liar paradox has turned into Tarski's th...

An implication of Gödel’s incompleteness theorem

A proof of Gödel’s incompleteness theorem is given. With this new proof a transfinite extension of Gödel’s theorem is considered. It is shown that if one assumes the set theory ZFC on the meta level as well as on the object level, a contradiction arises. The cause is shown to be the implicit identification of the meta level and the object level hidden behind the Gödel numbering. An implication ...

Gödel's Incompleteness Theorem

There are some topics in mathematics which, although enjoying a certain fame, have traditionally been considered either too complicated or of insufficient importance to be included in the core mathematical curriculum. Customarily, such subjects are relegated to optional units of the syllabus, independent study projects, seminar papers, and math club talks. Among such topics are several which re...

Redundancies in the Hilbert-Bernays Derivability Conditions for Gödel's Second Incompleteness Theorem