Is this because of the Löwenheim-Skolem theorem?

Question

I just started studying FOL. My book says this. Does this mean the Löwenheim-Skolem theorem? Before I read more, I'm asking this because it's so interesting for me. If you are interested, you can download this note here.

Answer 1

The Löwenheim-Skolem theorem offers an easy way to justify the assertion. But you can prove it in other ways.

For example, the theory of algebraically closed fields of characteristic 0 (ACF$_0$) is complete, so all models are elementarily equivalent. However, a countable model and an uncountable model cannot be isomorphic, of course. Likewise for the theory of dense linear orders without endpoints (DLO).

You don't need Löwenheim-Skolem to prove these completeness facts. One way is using quantifier elimination.

Alternately, the completeness of DLO follows from the fact that all countable models are isomorphic ($\aleph_0$-categoricity) and for ACF$_0$, from the fact that all models of a given uncountable cardinality are isomorphic ($\kappa$-categoricity for all $\kappa>\aleph_0$).

Of course, these use the implication that categoricity in any given cardinal implies completeness. Often one proves this using the downward LS theorem. But the usual proofs of Gödel completeness show that a consistent countable theory has a countable model, so you can get the implication for $\aleph_0$-categoricity that way. For $\kappa$-categoricity, just add enough constants. (However, it's worth noting that the usual proof of downward LS is pretty similar to the Henkin proof of Gödel completeness, and "add constants" plus Gödel completeness is how you prove upward LS.)

Delving a little deeper, we have examples of non-isomorphic countable models of ACF$_0$, and non-isomorphic models of DLO of the same uncountable cardinality. For ACF$_0$, you can take the rationals and adjoin either one or two transcendentals, and then take algebraic closures. For DLO, you have the reals vs. (say) the open interval (0,1) union the rationals in (1,2).

So DLO is $\aleph_0$-categorical, but not $\kappa$-categorical for any $\kappa>\aleph_0$; the first fact gives a way to prove that DLO is complete, i.e., all models are elementarily equivalent. Obversely, ACF$_0$ is $\kappa$-categorical for all $\kappa>\aleph_0$, but not $\aleph_0$-categorical, with the first fact providing a way to show the theory's completeness.

Finally, I feel obligated to mention Morley's theorem: a theory categorical in one uncountable cardinal is categorical in all uncountable cardinals. So there are only four possibilities vis-a-vis categoricity, and DLO and ACF$_0$ furnish examples of two of them.