I am currently studying Gödel's Incompleteness Theorems, and I have a question about $\omega$-inconsistency.
Some say that a theory $T$ is $ω$-inconsistent if $T$ proves $P(0)$, $P(1)$, $P(2)$,... and also proves that there is a number $N$ for which $P(N)$ fails.
Others instead say that $T$ is $\omega$-inconsistent if it proves that there is a number for which $P(N)$ holds, but proves that $P(0)$ fails, $P(1)$ fails,...
Which one is wrong? Or (as I think) are they equivalent?
Turning Rob Arthan's initial comment into an answer to kick this off the "unanswered" queue, and CW-ifying to avoid reputation gain for someone else's work:
The two definitions are equivalent: given one, we get the other by swapping $P$ and $\neg P$. (This reveals an omitted clause in the definitions: we need to begin with "for some $P$" There's no single $P$ which "tests for $\omega$-inconsistency.")