Extensional Independence

Joel Hamkins asks whether there is a ${\mathrm{\Pi }}_1^0$-formula $\rho (x)$ such that $\rho (\phi )$ is independent over $\mathsf{P}\mathsf{A}+\phi$, if this theory is consistent, where this construction is extensional in $\phi$ with respect to $\mathsf{P}\mathsf{A}$-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 $\rho$ to be ${\mathrm{\Pi }}_1^0$-conservative, if we ask for the still weaker Conditional Extensionality. We show that an intensional version of the result for Conditional Extensionality cannot work.