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Here is a construction of a theory T with the properties of Self-PA. That is, 1) T extends PA and 2) T can prove the consistency of T. Of course, by Godel's second incompleteness theorem, T must be inconsistent, but it is not obviously inconsistent.

In addition to the axioms of PA, T will have one additional axiom PHI, to be chosen presently.

By the devices (due to Godel) used to formalize "PA is consistent" in PA we can find a formula S(x) with one free variable, x, in the language of PA which expresses the following:

a) x is the Godel number of a ... (read more)