Upward Lowenheim Skolem use

Question

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?

Answer 1

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 $T$.)