Let’s fix a reasonable, sound axiomatic theory T in the language of first-order arithmetic. Gödel exhibited a sentence that T does not prove, namely Con(T). Though there are T-unprovable sentences that are strictly weaker than Con(T), it has been widely observed that none of them are “natural.” It is unclear how to make this observation mathematically precise, since “naturalness” is an informal notion.

We can sharpen the question somewhat by replacing talk of “natural sentences” with talk of operators on sentences with precisely defined invariance properties. We will present two results on such operators: First, a positive result suggesting that the consistency operator is the weakest natural way to uniformly extend axiomatic theories. Second, a negative result to the effect that the positive result does not generalize to iterates of the consistency operator.