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.