On the inevitability of the consistency operator

Abstract

We examine recursive monotonic functions on the Lindenbaum algebra of $\mathsf{EA}$. We prove that no such function sends every consistent $\phi$ to a sentence with deductive strength strictly between $\phi$ and $(\phi \wedge \mathsf{Con}(\phi ))$. We generalize this result to iterates of consistency into the effective transfinite. We then prove that for any recursive monotonic function $f$, if there is an iterate of $\mathsf{Con}$ that bounds $f$ everywhere, then $f$ must be somewhere equal to an iterate of $\mathsf{Con}$.

Cited by in corpus (1)

• On the hierarchy of natural theories