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I am asked to verify whether or not you can axiomatise the theory of countable dense total orderings in the language of partially ordered sets.

We have the Upward Lowenheim-Skolem Theorem, telling us that for a language $L$ and theory $S\subset L$ then if $S$ has an infinite model, it has an uncountable model.

My first thought was that if the theory of countable dense total orderings was axiomatisable, then $\mathbb Q$ with the usual ordering would an infinite model, hence by Upward Lowenheim-Skolem, there exists an uncountable model, and so we have a contradiction.

However, after more thought is this really a contradiction? Can't it be the case that "inside" of this uncountable model, it looks like its countable?

Is it perhaps the case then that the theory is indeed axiomatisable?

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  • $\begingroup$ Inside what model? What do you mean by "inside"? And by "it looks like"? $\endgroup$ – Andrés E. Caicedo May 21 '18 at 21:00
  • $\begingroup$ @AndrésE.Caicedo If we were to have an uncountable model of the theory of countable dense total orderings from the Upward Lowenheim-Skolem, then in what sense is it uncountable? $\endgroup$ – user366818 May 21 '18 at 21:02
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    $\begingroup$ In that it satisfies the definition of "uncountable". You need to explain yourself better. $\endgroup$ – Andrés E. Caicedo May 21 '18 at 21:05
  • $\begingroup$ @AndrésE.Caicedo okay a different question then, is there anything wrong with the existence of an uncountable model for the theory of countable dense total orderings? How can a model for something countable be uncountable? $\endgroup$ – user366818 May 21 '18 at 21:07
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    $\begingroup$ The theory of countable dense linear orderings is simply the set of all sentences true in all such structures. This theory has uncountable models. The question is whether the class of countable blahblah is axiomatizable, meaning whether there is a theory such that a structure is in the class iff it satisfies the theory. $\endgroup$ – Andrés E. Caicedo May 21 '18 at 21:13
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I think there's a bit of confusion about what "the theory of uncountable [things]" means.

"The theory of countable [things]" is just the set of sentences satisfied by every countable [thing]. There's no rule that it not also be satisfied by some uncountable [thing]: "the theory of countable [things]" is not required to hold of only the countable [things].

It is, however, true that the class of countable [things] is almost never an elementary class - that is, if there is even one infinite [thing], then there is no first-order theory $T$ which is true of exactly the countable [things].

That is:

The theory of a class of structures $\mathcal{K}$ is just $Th(\mathcal{K})=\{\varphi: \forall M\in\mathcal{K}, M\models\varphi\}.$ In general, the theory of a class of structures describes a broader class of structures than the original class: $Mod(Th(\mathcal{K}))\supsetneq \mathcal{K}$ in general. (Here "$Mod(T)$" denotes the class of models of $T$.)

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