How the Law of Excluded Middle Pertains to the Second Incompleteness Theorem and its Boundary-Case Exceptions

TL;DR

This paper explores how the Law of the Excluded Middle influences the Second Incompleteness Theorem, demonstrating that certain boundary-case exceptions collapse under classical logic assumptions in semantic tableau systems.

Contribution

It shows that boundary-case exceptions to the Second Incompleteness Theorem depend on treating the Law of the Excluded Middle as a schema rather than a derived theorem.

Findings

Boundary-case evasions collapse with classical logic treatment

Semantic tableau admits partial exceptions under specific formalism

Recognition of self-consistency affects theorem applicability

Abstract

Our earlier publications showed semantic tableau admits partial exceptions to the Second Incompleteness Theorem where a formalism recognizes its self consistency and views multiplication as a 3-way relation (rather than as a total function). We now show these boundary-case evasions will collapse if the Law of the Excluded Middle is treated by tableau as a schema of logical axioms (instead of as derived theorems).