Realisation of types in complete theory

Question

This is a homework question so I am not looking for an answer... I'm just really stuck and feel as though I'm missing a big point so I'd like a hint or a tip.

I'm trying to show that any 2 (complete) types over a complete theory $T$ can be realised by a common model.

I have trouble even getting started... Any help would be greatly appreciated!

Answer 1

Hint. Expand the language with two constant symbols, and consider the set of sentences asserting that the first constant realizes one of the types and the second constant realizes the other. Can you show this is finitely consistent?

Further hint:

Finite consistency means we need to show $\varphi(c_1) \wedge \psi(c_2)$ is consistent, where $\varphi(x)$ is an arbitrary formula from the first type and $\psi(y)$ is an arbitrary formula from the second (since the types are closed under conjunction). Note that $T \models (\exists x) \varphi(x)$ and $T \models (\exists y) \psi(y)$. (Why? Be careful: this is where you use that $T$ is complete.)