Extensional Independence

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 $\mathsf{PA}$-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 *Consistent Extensionality*. We also prove that we can demand the negation of $\ddot{I}$ to be ${\hat{I}}_1^0$-conservative, if we ask for the still weaker *Conditional Extensionality*. We show that an intensional version of the result for Conditional Extensionality cannot work.

This preprint extends and supersedes the earlier preprint [arXiv:2502.09109] *On a Question of Hamkins'* by Albert Visser