The thing is, aren't you using sets when you prove the Completeness Theorem? In the completeness theorem you are referring to theories which are sets of first order sentences, which themselves are made up of symbols from a set. Don't you need sets in the first place to do model theory? I mean, what are the foundations of model theory?

I simply mean the vector <p_x,p_y,p_z> which one uses in vector calculus. It is normal to the level set p=0 at every point.