Theory of Fields $\omega$-Inconsistent?

Question

A theory is $\omega$-inconsistent if there is a predicate $P(n)$ that is true for every standard natural number yet not true for all numbers. Consider the theory of finite fields and let $P(x) = (Sx \neq 0)$. In any pseudo-finite field, $P(x)$ is true for every standard natural number yet false for $-1$. Is the theory of finite fields $\omega$-inconsistent? What about the theory of real closed fields?

Answer 1

The expression $\forall n \varphi(n)$, where $n$ ranges over the natural numbers, can be formulated in neither theory. So there is no notion of $\omega$-inconsistency for either mentioned theory.