Refinements of provability and consistency principles for the second incompleteness theorem

Abstract

This paper continues the author's previous study \cite{Kura20}, showing that several weak principles inspired by non-normal modal logic suffice to derive various refined forms of the second incompleteness theorem. Among the main results of the present paper, we show that the set $\{\mathbf{E},\mathbf{C},D3\}$ suffices to establish the unprovability of the consistency statement $\neg {\mathrm{Pr}}_T(┌0=1┐)$. We also prove that the set $\{{\mathbf{E}}^{\mathrm{U}},\mathbf{C}{\mathbf{B}}_{\exists }\}$ yields formalized ${\Sigma }_1$-completeness.

24 pages