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I once asked whether the successor function can be defined in the structure $(\mathbb{N}, +, 0)$ without parameters. Now, I am asking whether the successor function can be defined in the structure $(\mathbb{Z}, +)$ without parameters. I left out $0$ in that structure because $0$ can certainly be defined without parameters in that structure.

user107952
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    How is this not trivially "no"? The sign-swapping automorphism doesn't preserve "successor", and there doesn't seem to be any internal way to tell them apart -- you need a parameter to label the sign. – wnoise Apr 23 '21 at 22:49
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    A more interesting result is the $\emptyset$-undefinability of the unit distance relation $U(a,b)\iff \vert a-b\vert=1$. This can't be established by automorphisms, since all (= both) automorphisms of $(\mathbb{Z},+)$ respect $U$. Moreover, since $U$ is definable by an infinite conjunction of universal sentences, we can't use EF-games either. We need a quantifier elimination argument, but $(\mathbb{Z},+)$ doesn't admit quantifier elimination so we first need to find the right expanded language. At a glance adding subtraction and a "mod $n$" unary predicate for each $n$ should do the job. – Noah Schweber Apr 23 '21 at 23:19
  • @AlexKruckman Hehehe :P. Do you know a better way to prove my comment than expanded-language-QE? – Noah Schweber Apr 24 '21 at 00:06
  • @NoahSchweber QE would be my go to approach anyway. IIRC, you're correct that adding 0,-, and a predicate for divisibility by p for each prime p gives QE. – Alex Kruckman Apr 24 '21 at 00:25
  • @wnoise You should turn your comment into an answer. – Noah Schweber Apr 27 '21 at 19:56
  • @NoahSchweber I'm afraid I should really leave it to someone who actually knows model theory. – wnoise Apr 27 '21 at 21:13
  • @wnoise I don't think so, you got it exactly right. Since it's not preserved by automorphisms, successor isn't definable in $(\mathbb{Z};+)$ according to any logic (in a precise sense), let alone first-order logic. As an answer, your comment doesn't need any elaboration. – Noah Schweber Apr 27 '21 at 21:15
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    But if you're not comfortable with it, I can post an answer crediting you if you'd prefer. – Noah Schweber Apr 27 '21 at 21:16
  • @NoahSchweber Yeah, I want you to post it as an answer. – user107952 May 01 '21 at 16:48
  • @user107952 OK, I'll do that in a bit. – Noah Schweber May 01 '21 at 17:22

1 Answers1

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This expands wnoise's comment above into an answer; I've made it CW to avoid reputation gain for their work, and if they post an answer of their own I'll delete this one.

The answer is no, the successor function is not definable in $\mathcal{Z}=(\mathbb{Z};+)$. There is a simple proof of this: the map $$\alpha: z\mapsto-z$$ is an automorphism of $\mathcal{Z}$ which does not preserve the successor operation, but all definable functions(/relations/elements) must be preserved by automorphisms.

Note that this only uses the isomorphism-invariance of first-order logic: if $$i:\mathcal{A}\rightarrow\mathcal{B}$$ is an isomorphism, $a_1,...,a_n\in\mathcal{A}$, and $\varphi(x_1,...,x_n)$ is a formula, then we must have $$\mathcal{A}\models\varphi(a_1,...,a_n)\quad\iff\quad \mathcal{B}\models\varphi(i(a_1),...,i(a_n)).$$ For first-order logic this can be proved by induction on the complexity of $\varphi$, but isomorphism invariance is usually taken as part of the$^1$ definition of a logic. Thus, we have:

There is no logic according to which the successor function is definable in $\mathcal{Z}$.

Of course there are limitations to the automorphism approach. For example, the "unit distance" relation is fixed by all automorphisms of $\mathcal{Z}$, and in fact is definable over $\mathcal{Z}$ in either of the logics $\mathcal{L}_{\omega_1,\omega}$ or $\mathsf{SOL}$ (infinitary and second-order logic, respectively). So to show that the unit distance relation is not first-order definable over $\mathcal{Z}$ we have to do some work.

$^1$OK, this isn't really accurate: there are many different definitions out there, so what I wrote doesn't really parse. However, all the various notions that I'm aware of agree on this. For what it's worth, the one I think about most is the notion of regular logic, for which see the end of Ebbinghaus/Flum/Thomas' Mathematical Logic or the beginning of the collection Model-theoretic logics.

Noah Schweber
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