ON THE INEVITABILITY OF THE CONSISTENCY OPERATOR

We examine recursive monotonic functions on the Lindenbaum algebra of EA. We prove that no such function sends every consistent phi to a sentence with deductive strength strictly between phi and (phi Lambda 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 Con that bounds f everywhere, then f must be somewhere equal to an iterate of Con.