On a Question of Hamkins'

Joel Hamkins asks whether there is a $\Pi^0_1$-formula $\rho(x)$ such that $\rho({\ulcorner \phi \urcorner})$ is independent over ${\sf PA}+\phi$, if this theory is consistent, where this construction is extensional in $\phi$ 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 $\Pi^0_1$-formula $\rho(x)$ that is extensional and $\Pi^0_1$-flexible. We leave one important question open: what happens when we weaken Extensionality to Consistent Extensionality, i.e., Extensionality for consistent extensions?