Let $T$ be a theory which is consistent and $\omega$-inconsistent, how to prove that $T$ is $\omega$-incomplete?
Recall: A theory $T$ is $\omega$-incomplete if there is a formula $A$ with only one free variable such that the theoy $T$ proves $A(\bar{n})$ for all natural number $n$ but it doesn't prove $\forall x A(x)$. ($\bar{n}$ is the numeral of $n$)
