A classification of incompleteness statements
Henry Towsner, James Walsh · Canadian Mathematical Bulletin · 2025
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 Gödel’s second incompleteness theorem that hold within second-order arithmetic.