On a Question of Hamkins'

Abstract

Joel Hamkins asks whether there is a ${\hat{I}}_1^0$-formula $\ddot{I}(x)$ such that $\ddot{I}(┌\ddot{I}•┐)$ is independent over $\mathsf{PA}+\ddot{I}•$, if this theory is consistent, where this construction is extensional in $\ddot{I}•$ with respect to {\sf PA}-provable equivalence. We show that there can be no such extensional Rosser formula of any complexity. We give a positive answer to Hamkins' question for the case where we replace Extensionality by a weaker demand that we call \emph{Conditional Extensionality}. For this case, we prove an even stronger result, to wit, there is a ${\hat{I}}_1^0$-formula $\ddot{I}(x)$ that is extensional and ${\hat{I}}_1^0$-flexible. We leave one important question open: what happens when we weaken Extensionality to Consistent Extensionality, i.e., Extensionality for consistent extensions?

This preprint has been superseded by preprint ArXiv:2506.13524, Extensional Independence, by Taishi Kurahashi and Albert Visser