Does an infinite axiom system exist for a finitely axiomatizable theory in first order logic?

Question

What I'm trying to find:

A concrete example in predicate logic with the following characteristics:

1. An infinite system of axioms for a finitely axiomatizable theory.
2. Any finite subset of the infinite axiom system should no longer axiomatize the theory.

My first intuition was to recursivly define a infinite set of formulas which axiomatize the theory, but I didn't manage to build any examples myself. So I'm not sure if such a axiomatic system is even possible

Answer 1

This is impossible, by the compactness theorem. See my answer here: https://math.stackexchange.com/a/3814809/7062