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Let $L$ be a first order language and $A$ be an L-structure. Let $\sigma$ be an L-sentence.

In most of the mathematics I have seen a structure is defined by the axioms it satisfies. These may or may not be expressible as first order sentences. For instance groups have a first order axiomatisation where as the axioms of a topological space are not entirely first order. So I would say $A \models \sigma$ just when there is a proof (not necessarily a formal first order proof) of $\sigma$ from the axioms of $A$. So to me a structure always comes with a set of axioms it satisfies.

Somehow in the model theory I have read the situation seems to be inverted. We are given a structure and are asking whether we can find a (first order) axiomatisation for it? But how then is a structure defined, if not via some kind of axiomatisation? In order to even talk about a structure it seems to me one has to have some kind of axiomatisation already in mind. Concretely, I believe, Gödel showed $Th(\mathbb{N})$ isn't (first order) axiomatizable. But how, if not by some kind of axioms (e.g. PA), is $\mathbb{N}$ defined?

The only way I see out of this is that structures are defined by axioms ( maybe not first order expressible) and model theorists ask when a first order axiomatisation exists. Is this the full story?

Does anyone see what I am missing? Many thanks!

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1 Answer 1

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In most of the mathematics I have seen a structure is defined by the axioms it satisfies.

I'm going to go ahead and disagree with this, at least to a certain extent. Consider the following two approaches to "defining the reals:"

  1. As the unique complete ordered field.

  2. Via equivalence classes of Cauchy sequences.

Each of these is done with decent frequency; however, only the former is actually isomorphism invariant, and so only the former should be thought of in terms of axioms (first-order or otherwise). Under the hood, both approaches amount to proving an existence-and-appropriate-uniqueness result in an appropriate background theory, although they demand different amounts of uniqueness.

Even before we get to "first-order vs. other," model theory raises the distinction between isomorphism-invariant properties and constructions which require auxiliary work. This doesn't have to be a normative distinction - I find "auxiliary work" quite interesting - but it is a valuable one in many ways.


OK, so what about $\mathbb{N}$? (Let's work in $\mathsf{ZFC}$ for simplicity.)

Well, first of all note that we again have two very different ways of "defining $\mathbb{N}$:" we could prove (say) that there is a unique-up-to-isomorphism prime model of $\mathsf{PA}$, or we could explicitly define additive and multiplicative structure on $\omega$. In either case, we get a theorem which says "$\mathbb{N}$ exists." Moreover, we can prove auxiliary results connecting the different versions of this theorem, so that things are reasonably copacetic.

Godel's incompleteness theorem does put limits on how we could hope to "characterize $\mathbb{N}$," but it doesn't in any way stand in tension with the foregoing.

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  • $\begingroup$ Thank you for your answer. The issue I have is that it seems non-sensical to want to find axioms for a structure already defined by certain axioms. If I have understood your answer correctly, you are suggesting that my premise is flawed, i.e. that structures are not in general defined via axioms. Somehow I am not totally convinced, as for example it seems to me that the axiomatic definition is often the most convenient (and in those cases my point still stands)(e.g. groups; finding an axiomatisation for the theory of groups seems non-sensical, as groups are already defined by axioms.) $\endgroup$
    – user
    Jan 1 at 8:41
  • $\begingroup$ @user Having defined a structure (such as $\mathbb{N}$) or class of structures (such as the class of groups) in any way at all, we can then ask: is there a definition of this structure or class of structures via axioms of such-and-such a form (e.g. first-order, equational, or even just "shorter")? E.g. even up to isomorphism $(\mathbb{N};+,\times)$ can't be pinned down "from the inside" by first-order axioms, due to the compactness theorem. A particular instance of this may or may not be interesting, but I don't see how it can be nonsensical. $\endgroup$ Jan 1 at 8:44
  • $\begingroup$ As you put things in your comment I am completely convinced ( thanks :) ), i.e. asking for a first order axiomatisation of a structure ( however defined) is a reasonable and consistent thing to do. Not at all disagreeing with your point: Would you agree that in the case of structures already defined via a first order axiomatisation this however becomes rather redundant/ trivial to find such an axiomatisation? $\endgroup$
    – user
    Jan 1 at 9:01
  • $\begingroup$ @user Sure, but I wonder if you're conflating exposition with results here. It's quite reasonable to say "the class of groups is an example of a first-order-axiomatizable class" to help students connect a new idea (axiomatizable classes) with some concrete examples. Maybe you can give an example of some of the model-theoretic language you're objecting to? $\endgroup$ Jan 1 at 9:04
  • $\begingroup$ Thanks, you've perfectly cleared things up for me and with regards to your last question I'm not sure I have any more issues. ( The question in my previous comment was not a criticism of anything, but rather just to check my understanding) $\endgroup$
    – user
    Jan 1 at 9:09

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