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...

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

Gödel's Incompleteness Theorems: A Revolutionary View of the Nature of Mathematical Pursuits

The work of the mathematician Kurt Gödel changed the face of mathematics forever. His famous incompleteness theorem proved that any formalized system of mathematics would always contain statements that were undecidable, showing that there are certain inherent limitations to the way many mathematicians studies mathematics. This paper provides a history of the mathematical developments that laid ...

Gödel's Incompleteness Theorems

Gödel’s two incompleteness theorems [2] are formalised, following a careful presentation by Świerczkowski [3], in the theory of hereditarily finite sets. This represents the first ever machine-assisted proof of the second incompleteness theorem. Compared with traditional formalisations using Peano arithmetic [1], coding is simpler, with no need to formalise the notion of multiplication (let alo...

Some Transfinite Generalisations of Gödel's Incompleteness Theorem

A Machine-Assisted Proof of Gödel's Incompleteness theorems for the Theory of Hereditarily Finite Sets

A formalization of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows Świerczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that...