I know that by Tennenbaum's Theorem, non-standard models of arithmetic "don't know" which of their elements form a standard model of arithmetic. However, often facts that are opaque to the model from an "internal" perspective can be recognized "from the outside". (After all, if this weren't possible we wouldn't know that these models were non-standard.) For example, countable models of set theory can be seen to be "non-standard", in a sense, because they classify certain sets as uncountable when "from the outside" we can see this is simply a weird side-effect of that model lacking an injective function from these sets into their natural numbers.

So, suppose that from an external perspective we know which substructure of a non-standard model of arithmetic contains the standard numbers (even though the model can't identify this substructure from its perspective) -- perhaps we started with a standard model and constructed the non-standard model from it.

Could the construction of the non-standard model add to the properties of the standard numbers we started with? Or does the construction of the non-standard model always add completely disconnected structure to the initial structure, such that the properties of the initial structure remain completely unchanged?

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    $\begingroup$ What does "add to the properties" mean here? $\endgroup$ – Mees de Vries Jun 21 '17 at 21:24
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    $\begingroup$ It would help to be more precise in your question, although some of these details may be obvious to specialists. Are you asking, for instance: is the inclusion of the standard numbers into any model of first-order Peano arithmetic an elementary embedding? $\endgroup$ – Kevin Arlin Jun 21 '17 at 21:25
  • $\begingroup$ @KevinCarlson, no, it is not: consider the theory $\mathrm{PA} + \neg \mathrm{Con}(\mathrm{PA})$. A model of this (consistent) theory is a non-standard model of PA. Edit: my bad, I misread your comment. But here is the answer anyway. 0:-) $\endgroup$ – Mees de Vries Jun 21 '17 at 21:26
  • $\begingroup$ @MeesdeVries I was kind of leaving that intentionally vague because I was curious what interesting precisifications (if any) there were of that vague phrase. I know that the non-standard model would have to conservatively extend the original model so as to remain a model of arithmetic. But I wasn't sure what, if any, new properties might become expressible that give us, say, new (and not totally uninterestingly gerrymandered) open formulae satisfied by the "standard numbers" we started with. I would definitely be grateful for other suggestions for how to precisify my vague phrase. $\endgroup$ – Dennis Jun 21 '17 at 21:33
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    $\begingroup$ @KevinCarlson Noah did a great job of responding to my muddled thoughts. With the help of his answer, I think I can say my question was: is there any formula $\phi(x)$ (with $x$ not a dummy variable) that isn't a part of true arithmetic, but which is satisfied by a number in the "standard" substructure of a non-standard model of arithmetic. The intuitive question I had that I was trying to make precise is whether the non-standard models of PA "muck about" with their standard substructure in some way that doesn't change any truth values of sentences in true arithmetic (obviously), but shows... $\endgroup$ – Dennis Jun 21 '17 at 22:34

Since each element of $\mathbb{N}$ is definable, if $M$ is a nonstandard model of true arithmetic then the inclusion map is elementary. So the standard elements satisfy all the same sentences they did originally.

If instead of true arithmetic we are looking at PA (or similar), then even $\Pi^0_1$ facts need not be preserved: e.g. consider the formula $\varphi(x)\equiv$"PA is inconsistent" (note that $x$ is just a dummy variable here). If $M$ is a model of PA + $\neg$Con(PA), then $\varphi(0)$ holds in $M$ but fails in $\mathbb{N}$; and $\varphi$ is $\Pi^0_1$.

Conversely, if $M$ is any $\{+, \times, <\}$-structure with a unique initial segment isomorphic to $\mathbb{N}$, let alone a nonstandard model of PA, then - via that isomorphism - $\mathbb{N}$ embeds in $M$ in a $\Sigma^0_1$-preserving way. So we can never make a true $\Sigma^0_1$ fact become false in a nonstandard model.

A less trivial example of a $\Pi^0_1$ formula whose truth value on a standard number can change from "true" to "false" when we pass to a nonstandard model: let $\varphi(n)$ be "the sentence with Godel number $n$ is consistent with PA." Note that the reason for this formula "fluctuating" is the same as the above, it's just that here we don't have a dummy variable.

In fact, since each standard natural is definable, we are really just talking about sentences changing truth value.

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  • $\begingroup$ Great! I was in the process of responding to your comment when you did a wonderful job of unmuddling my thoughts for me! Is there any (obviously non-$\Sigma^0_1$) formula $\phi(x)$ that could be made true in a non-standard model of $\mathsf{PA}$ but which isn't part of true arithmetic, and where $x$ isn't a dummy variable? (The worry is that having $x$ as a dummy variable makes it hard to see $\phi(0)$ as predicating any property to 0.) $\endgroup$ – Dennis Jun 21 '17 at 22:24
  • $\begingroup$ Thanks for the edit in response to my comment! The last part expanding upon the impact of definability on predication -- that it essentially collapses satisfaction of open arithmetical formulas to satisfaction of a sentence -- is particularly helpful! $\endgroup$ – Dennis Jun 22 '17 at 0:02

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