[FOM] Is PA + ~Con(PA) a complete theory?

[Andrew Polonsky]

Let T be the theory obtained by adding to PA the axiom

Incon(PA) = exists n. n is a code of a PA-derivation of Falsum

Since T is a consistent c.e. theory extending PA, one would expect to have
undecidable propositions in it.  Are there any known examples of such
propositions?

(The obvious candidate might be
Con(PA + Incon(PA))
However, the negation of the above statement can be derived from an axiom.)

Cheers,
Andrew
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