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You Provably Can't Trust Yourself

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 sentence, S, of the language of PA;

b) The theory "PA + S" is consistent.

By Godel's self-referential lemma, there is a sentence PHI, with Godel number q such that PA proves:

PHI if and only if S([q]).

(Loosely speaking, PHI says the theory obtained by adjoining me to PA is consistent.)

If we take T to be the theory "PA + PHI" then T has the properties stated at the start of this posting.

Of course, to fully understand the argument just given one needs some familiarity with mathematical logic. Enderton's text "A Mathematical Introduction to Logic" covers all the necessary background material.