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I've been diving into Gödel's Incompleteness Theorems lately and it seems like all the weird scenarios I've encountered hinges on the existence of non-standard models of PA.

For example, if we have a consistent system S that is strong enough to express arithmetic, the statement Cons(S) = "there is no Gödel number $x$ that encodes a proof of 0 = 1" is independent of S itself. That means we can a create model where $\neg$Cons(S) is true. This is possible because in these models, there will exist a non-standard natural number that seemingly encodes a proof of 0=1.

Another example involves the Goldbach Conjecture (same logic applies to the Riemann Hypothesis). If we can prove that Goldbach is independent of PA (an unlikely scenario but let's just say that it's true), then it seems like this means Goldbach is true because we would be able to find a counterexample (i.e. an even number greater than 2 that's not the sum of two primes) in a finite number of steps if Goldbach is false. The reason why Goldbach could be independent of PA is because there are non-standard models of PA where there are non-standard naturals for which Goldbach is false.

These examples lead me to the question: what is the standard model of PA? Is it the von Neumann ordinals, given by the axiom of infinity? How do we decide which model is standard and which is non-standard? Is there actually a collection of standard models, which are equivalent up to an isomorphism?

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    $\begingroup$ See also Non-standard model of arithmetic $\endgroup$ Commented Nov 2, 2021 at 14:18
  • $\begingroup$ this is a difficult question, but just a little remark: von Neumann model (and theorems valid in it) of natural numbers depends on the model of (say) ZFC within which you construct it. So it cannot be quite used as the "standard model of natural numbers", unless you somehow fix a model of ZFC, which seems even more difficult. $\endgroup$
    – user8268
    Commented Nov 2, 2021 at 14:21
  • $\begingroup$ @user8268 You comment mixes up "levels" in a misleading way. Working in ZFC (i.e., with ZFC as our metatheory), "the natural numbers" refers to a specific and unique structure, namely the set $\omega$ of finite ordinals. There is no need to fix a model $M$ of ZFC and look at the "natural numbers" $\omega^M$ internal to that model, unless I want to study ZFC (as an object theory). And even if I do that, and I find lots of different models $\omega^{M'}$ for different models $M'$ of ZFC, I still have the "real" natural numbers $\omega$ in the background. $\endgroup$ Commented Nov 2, 2021 at 14:46

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It isn't the naturals which have a standard model, it is first-order PA (for example) which has a standard model, indeed the standard model, and the domain of the standard model is the natural numbers.

Thus Kaye, for example, in his classic book on models of PA:

The structure $\mathbb{N}$ (called the standard model) is the $\mathcal{L}_A$ structure whose domain is the set of non-negative integers, {0,1,2,3, ... }, and where the symbols in $\mathcal{L}_A$ [the language of first-order arithmetic] are given their obvious interpretation.

You can cook up models of PA that are isomorphic to this one. But we would not say that an isomorphic model whose domain consists of all ordered pairs of the form <Donald Trump, $n$> is standard, as e.g. the ordered pair of Trump and twenty is not itself a number, and the standard model is built out of numbers!

If you want to know about standard vs non standard models of PA, you can't do better than to dip into the opening chapters of Kaye's wonderful book.

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  • $\begingroup$ Thanks for the clarification about models. I was a bit imprecise with my language. I've edited my question so that other people reading it wouldn't be confused by it. $\endgroup$
    – ghost
    Commented Nov 2, 2021 at 14:35

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