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What is a simple (the simplest?) axiom which can be added to the usual PA axioms so that the new "non-standard PA theory" no longer has the Standard Model as one of its models? Assuming of course that the new axiom is consistent with the original axioms.

EDIT: I intend a first order theory. If possible by adding a single new axiom. If absolutely necessary, then an RE axiom schema.

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    $\begingroup$ You mean adding a constant $\gamma$ and infinitely many axioms $\gamma\neq 0$, $\gamma\neq 1$, ... $\endgroup$
    – user10354138
    Commented Jul 18, 2020 at 11:38
  • $\begingroup$ I see what you are saying, and it would do the job alright. Except it would obviously be more than one axiom. Is it clear that we cannot do better? $\endgroup$
    – Tommy R. Jensen
    Commented Jul 18, 2020 at 12:14
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    $\begingroup$ It's not very simple, but does $\lnot \mathrm{Con}(\mathsf{PA})$ work? $\endgroup$
    – Nate Eldredge
    Commented Jul 18, 2020 at 14:32
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    $\begingroup$ @TommyR.Jensen: And $\lnot \mathrm{Con}(\mathsf{PA})$ is consistent with $\mathsf{PA}$, assuming $\mathsf{PA}$ is itself consistent. If it were not, then $\mathsf{PA}$ would prove $\mathrm{Con}(\mathsf{PA})$, which by Godel's second incompleteness theorem it cannot do. $\endgroup$
    – Nate Eldredge
    Commented Jul 18, 2020 at 16:12
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    $\begingroup$ @NateEldredge I was talking about a comment that seems to have disappeared, where it was asserted that PA plus the one axiom "there exists a non-zero non-asuccessor" added had the required property. $\endgroup$
    – David C. Ullrich
    Commented Jul 18, 2020 at 21:43

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Since the new axiom must be consistent but not hold in the standard model, it must be undecidable in $\sf PA$. But for most (all?) of the statements which we have proved undecidable in $\sf PA$, we know whether or not they hold in the standard model. We can therefore obtain a nonstandard theory by adding the statement or its negation to $\sf PA$.

So answering the question simply requires us to find the simplest statement which is known to be undecidable in $\sf PA$. Of course Gödel sentence was the first known example of such, but since then various simpler undecidable sentences have been found. The most famous are Goodstein's theorem (proved undecidable by Kirby and Paris), and the Strengthened Finite Ramsey Theorem (proved undecidable by Paris and Harrington). Both these statements are known to be true in the standard model, so we obtain a non-standard theory by adding the negation of one of them as an axiom.

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