Editing Gödel's incompleteness theorems (section)

=== Expressing consistency ===

There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of {{mvar|F}} as a formula in the language of {{mvar|F}}. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons({{mvar|F}}) from the second incompleteness theorem is a particular expression of consistency.

Other formalizations of the claim that {{mvar|F}} is consistent may be inequivalent in {{mvar|F}}, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent [[subset]] of PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration.)