Abstract:
It is a well-known empirical phenomenon that natural axiomatic theories are well-ordered by their consistency strength. To investigate this phenomenon, we examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent $\varphi$ to a sentence with deductive strength strictly between $\varphi$ and $(\varphi\land \mathrm{Con}(\varphi))$. 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 $\mathrm{Con}$ that bounds $f$ everywhere, then $f$ must be somewhere equal to an iterate of $\mathrm{Con}$.
