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:
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)