In propositional logic, one can prove the compactness theorem by extending a finitely satisfiable set of formulas X to maximal set Y, such that every formula or its negation is a member of Y. One then constructs a truth assignment, assigning a value of true to precisely the formulas that are members of Y.
Does this approach for proving the compactness theorem generalize to first order logic? Specifically, X can be extended while preserving finite satisfiability to include every formula or its negation. Then, a model is constructed by assigning every term to its equivalence class, where terms are in the same equivalence class if and only if the corresponding formula that asserts the terms’ equality appears in Y. The function symbols and relation symbols are interpreted naturally as well, but must be checked to ensure that the definitions are well defined. In my estimation, this results in a model for Y, but I am suspicious about the approach, since I don’t see any references for this proof approach. Is it possible that some formulas in Y with universal quantifiers are not true in this model?
