On the Invariance of G\"odel's Second Theorem with regard to Numberings

TL;DR

This paper investigates how G"odel's Second Theorem depends on the choice of numberings, showing that its core interpretation remains invariant under natural classes of formalizations despite certain counterexamples.

Contribution

It introduces deviant numberings that challenge traditional views but demonstrates invariance of the theorem's interpretation within natural classes of numberings.

Findings

Deviant numberings can produce provability predicates satisfying L"ob's conditions.

Counterexamples do not refute the theorem's interpretation when considering natural admissible numberings.

Invariance of G"odel's Second Theorem holds within a natural class of formalizations.

Abstract

The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of the underlying mathematical theorem depends on several arbitrary formalisation choices. In this paper I examine the theorem's dependency regarding G\"odel numberings. I introduce deviant numberings, yielding provability predicates satisfying L\"ob's conditions, which result in provable consistency sentences. According to the main result of this paper however, these "counterexamples" do not refute the theorem's prevalent interpretation, since once a natural class of admissible numberings is singled out, invariance is maintained.