In First Order Logic a formula is valid if every possible interpretation is a model for the formula, that is, make the formula true.
"Every possibile interpretation" means a infinite number of interpretation with any possible domain and any possible costant and function assignment ?
Yes. For a given first-order language, any set of objects is allowed as the domain of quantification; any objects from the domain are allowed as denotations of names of the language; any functions from objects of the domain to objects of the domain are allowed as interpretations of function-symbols; etc. ...
Which is of course why there is in general no hope of doing a brute-force search through possible interpretations to establish the validity of a wff of a first-order language in the way we can do a brute-force truth-table search through possible valuations of the atoms of a propositional language to establish validity of a propositional wff!