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My question is: is $\mathbb{N}$ a nonstandard model of arithmetic for someone else?

Let $\mathbb{N}^*$ be a nonstandard model of Peano Arithmetic. Then that consists of a copy of $\mathbb{N}$ followed by a dense linear order of copies of $\mathbb{Z}$.

Is the reverse also true? Is our familiar $\mathbb{N}$ nonstandard with respect to someone else?

If this question is ill-posed, it is because I do not fully understand the nuances of model theory, not to mention non-standard model theory. So feel free to re-pose this precisely and answer it.

I would like to understand whether some models of arithmetic are inherently richer than others, or whether they are all equally rich.

kdog
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  • "someone else" who ? Some "alien world" math ? – Mauro ALLEGRANZA May 14 '17 at 14:12
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    The description "standard model" is not without problems; see this post for history and this post for discussion. In a nutshell, the "st model" (if any) is so with regard to the "practical consensus" that we practice every day when we pay with our credit card, when we take a look at the clock, and many many other examples of the all-pervading language-game of counting. – Mauro ALLEGRANZA May 14 '17 at 14:51
  • Assuming PA is defined in the metalanguage of ZFC, maybe there is another metalanguage ZFC' and another PA' and some sort of natural mapping between models of PA and of PA' such that $\mathbb{N}^*$ is "equivalent" to numbers in ZFC', and $\mathbb{N}$ is "equivalent" to a nonstandard model in ZFC'? – kdog May 14 '17 at 14:57

2 Answers2

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Your question is not ill-posed and in fact admits a rather precise answer in the context of Joel David Hamkins' multiverse. The technical details of this may be beyond the level of this question but the basic assumption, on this view, is that there is not a single set-theoretic universe (governed by ZFC or whatever) but rather a multitude of such universes, in precise relationship to each other governed by the rules set out by Hamkins and collaborators.

On this view, whichever universe a particular observer happens to be in, his particular version of $\mathbb N$ is indeed nonstandard with respect to another universe in the Hamkins multiverse.

This recent MO thread provides additional information relevant to this question.

For a discussion see this paper.

Mikhail Katz
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The natural numbers are the prime model of PA: in a precise sense, they are the least rich model. There are lots of notions of richness of models; these are usually expressed in terms of the realized types, essentially what sort of behavior occurs in the model. E.g. the natural numbers are not very rich since "is divisible by $2$ and $3$ and $5$ and ..." is a very easily-describable behavior which is not realized in $\mathbb{N}$.

As to whether $\mathbb{N}$ is nonstandard in any sense: certainly not in the sense that we use the word when speaking of nonstandard models. Since $\mathbb{N}$ is the smallest model of PA, there's no way to shrink it further and still have a model of PA.

Incidentally, there's an interesting set-theoretic difficulty here: conceivably the thing we think is $\mathbb{N}$ could actually be a very clever nonstandard model. This has been investigated by a number of mathematicians and philosophers, and has had lots of discussion on this site (see e.g. this question). But that wouldn't mean "$\mathbb{N}$ is nonstandard," that would mean "we were wrong about what $\mathbb{N}$ is" - $\mathbb{N}$ is defined as the smallest thing it could be (this can be made completely precise via second-order logic (this question says a bit about that), but since second-order logic is intimately bound up in set theory this isn't entirely satisfying).

Noah Schweber
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  • I'm reading through some of these references. I am still not sure what the framework is for comparison of different models of natural number is, particularly if they derive from different ZFC metalanguages. Consider the number 1000. 1000 is not a nonstandard number right? Not in any model. But there is some mode, where there is some nonstandard number out there, that is not a number in my model. Say 1000. Would beings from that nonstandard universe think that 1000 was a number that like 1000 would be an element of any model of PA? – kdog May 15 '17 at 18:51
  • I guess I'm looking for an internal test. Like, in geometry, a being on a surface can perform measurements on that surface to distinguish it from say a surface of positive curvature, say by measuring the sum of angles of triangles. Is there something like that in logic? – kdog May 15 '17 at 18:54
  • @kdog: There is the notion of elementary equivalence which is diametrically opposed to what you're hoping for -- you can have nonisomorphic models for which it is impossible to distinguish them via the truth of any proposition in language of the theory. –  Oct 02 '17 at 21:51