ON THE INEVITABILITY OF THE CONSISTENCY OPERATOR

Abstract

AbstractWe examine recursive monotonic functions on the Lindenbaum algebra of $EA$. We prove that no such function sends every consistent φ to a sentence with deductive strength strictly between φ and $\left( {\varphi \wedge Con\left( \varphi \right)} \right)$. 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 $Con$.