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K is said to be a true theory if all proper of K are true in the standard model. Please show that there is an ω-consistent extension K of PA such that K is not a true theory. I think this is a strange assertion which states we can extend PA to a theory in which it contains all theorem of PA, but it has a proper axiom such that it i not true in standard model. Which axiom we can add to PA to provide this condition ?

  • A "typical" example of formula which is not true in the standard model is the negation of the Gödel-formula (see G's First Incompleteness Th). G's Theorem prove that the G-formula is true in the standard model but unprovable; thus it is not true in some non-standard model. Its negation is false in the standard model but also unprovable; thus it must be true in some non-standard model. – Mauro ALLEGRANZA Jun 15 '14 at 09:38
  • But please, note that $PA + (\lnot \gamma)$, where $\gamma$ is the G-formula, is $\omega$-inconsistet. Thus, it is not the answer to your question ... – Mauro ALLEGRANZA Jun 15 '14 at 10:17
  • @MauroALLEGRANZA - thank you for your attention. Why the theory PA + ¬ω-Con(PA) is ω-consistent? –  Jun 15 '14 at 14:26
  • The wikipedia article that was linked above has an explanation... @Mauro ALLEGRANZA - why not post that helpful link and comment as an answer? – Carl Mummert Jun 15 '14 at 18:01
  • " if all proper of K " what did you mean to write here instead of proper? – Andrés E. Caicedo Aug 30 '14 at 15:07

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See in Wiki the entry on ω-consistent theory.

The para regarding Arithmetically unsound, ω-consistent theories says :

Let ω-Con(PA) be the arithmetical sentence formalizing the statement "PA is ω-consistent". Then the theory PA + ¬ω-Con(PA) is unsound, but ω-consistent.

The argument is similar to the first example: a suitable version of the Hilbert-Bernays-Löb derivability conditions holds for the "provability predicate" ω-Prov(A) = ¬ω-Con(PA + ¬A), hence it satisfies an analogue of Gödel's second incompleteness theorem.

The first example referenced regards Consistent, ω-inconsistent theories :

Write PA for the theory Peano arithmetic, and Con(PA) for the statement of arithmetic that formalizes the claim "PA is consistent". Con(PA) could be of the form "For every natural number n, n is not the Gödel number of a proof from PA that 0=1". (This formulation uses 0=1 instead of a direct contradiction; that gives the same result, because PA certainly proves ¬0=1, so if it proved 0=1 as well we would have a contradiction, and on the other hand, if PA proves a contradiction, then it proves anything, including 0=1.)

Now, assuming PA is really consistent, it follows that PA + ¬Con(PA) is also consistent, for if it were not, then PA would prove Con(PA) (since an inconsistent theory proves every sentence), contradicting Gödel's second incompleteness theorem. However, PA + ¬Con(PA) is not ω-consistent. This is because, for any particular natural number n, PA + ¬Con(PA) proves that n is not the Gödel number of a proof that 0=1 (PA itself proves that fact; the extra assumption ¬Con(PA) is not needed). However, PA + ¬Con(PA) proves that, for some natural number n, n is the Gödel number of such a proof (this is just a direct restatement of the claim ¬Con(PA) ).

The argument appies to the theory K of the question because it is assumed that K is a true theory, i.e. all proper axioms of K are true in the standard model (we can say also that K is sound).

Being so, K is obviously ω-consistent and consistent.

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