Show that even though there are uncountably many real numbers, the Theory of (R) has a countable model

Question

So I've been contemplating this question and I'm not sure if I'm overthinking it or not. Tell me if this is accurate: To prove that Th(R) (Theory of Reals: all true sentences in the language <0,1,+,X,less than or equal to>) has a countable model, I first need to prove that a model exists(so I need to prove the this via compactness?), following that, I simply apply the Lowenheim-Skolem theorem in order to prove that Th(R) has a countable model. Is this correct? Or am I missing something here?

Any help is much appreciated.

Answer 1

Basically you are right, with a few remarks. You don't need to prove that a model of $Th(\mathbb {R})$ exists. By definition $Th(\mathbb {R})$ is the set of all true first order sentences in the standard language that hold in the model $\mathbb {R}$. Next, you notice that the standard language is itself countable and thus, applying Lowenheim-Skolem, you conclude that a countable model exists.