A classification of incompleteness statements

TL;DR

This paper characterizes when certain second-order arithmetic theories cannot prove their own soundness, thereby clarifying the limits of G"odel's second incompleteness theorem in this context.

Contribution

It provides a complete classification of incompleteness statements based on definability and soundness conditions in second-order arithmetic.

Findings

Identifies which combinations of definability levels prevent theories from proving their own soundness.

Delimits the scope of G"odel's second incompleteness theorem within second-order arithmetic.

Establishes a comprehensive framework for understanding incompleteness statements in this setting.

Abstract

For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G\"odel's second incompleteness theorem that hold within second-order arithmetic.