A classification of incompleteness statements

Abstract

For which choices of $X,Y,Z\in \{\hat{I}{\pounds }_1^1,{\hat{I}}_1^1\}$ 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 Gödel's second incompleteness theorem that hold within second-order arithmetic.