On a Question of Hamkins'

TL;DR

This paper investigates the existence of certain independent formulas over PA, proving non-existence under strict conditions and providing positive results under weaker assumptions, thus advancing understanding of independence and extensionality in formal theories.

Contribution

It demonstrates the non-existence of extensional Rosser formulas of any complexity and constructs extensional, flexible formulas under weaker conditions, addressing Hamkins' question.

Findings

No extensional Rosser formulas of any complexity exist.

Existential formulas with weaker 'Conditional Extensionality' are constructed.

The paper leaves open the case of 'Consistent Extensionality'.

Abstract

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?