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

Oosten, J. van J.vanOosten at uu.nl
Wed Jun 5 02:52:33 EDT 2013

The Gödel sentence for an arithmetical theory T requires, for the proof that it is independent, that T prove only true Sigma _1 formulas. Of course, PA+Incon(PA) lacks this property.
The Rosser sentence for T requires only that T be consistent.
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] on behalf of Andrew Polonsky [andrew.polonsky at gmail.com]
Sent: Tuesday, June 04, 2013 2:46 PM
To: fom at cs.nyu.edu
Subject: [FOM] Is PA + ~Con(PA) a complete theory?

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.)

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