On the inevitability of the consistency operator

TL;DR

This paper investigates the properties of recursive monotonic functions on the Lindenbaum algebra of elementary arithmetic, showing limitations on how they can relate to consistency statements and their iterates.

Contribution

It proves that no recursive monotonic function can consistently assign intermediate strength sentences between a statement and its consistency, and characterizes functions bounded by iterates of the consistency operator.

Findings

No recursive monotonic function sends all consistent formulas to intermediate strength sentences.

Iterates of the consistency operator effectively bound such functions only if they coincide with an iterate of consistency.

The results extend to effective transfinite iterations 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 $\varphi$ to a sentence with deductive strength strictly between $\varphi$ and $(\varphi\wedge\mathsf{Con}(\varphi))$. 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 $\mathsf{Con}$.