Does '$\omega$-consistency' mean to be consistent with $\Omega$? In a strong enough meta-logic such as ZFC or CC, for any $L_{\Omega}$-theory $T$, is $\Omega \models T$ equivalent to that $T$ is $\omega$-consistent?
Here, $L_{\Omega}$ is a first-order language whose signature is $\left\langle + , \cdot , S , 0 , < \right\rangle$, and $\Omega$ is the standard model of $L_{\Omega}$.
 A: No. This is explained (but not very clearly, I think) on the Wikipedia page for $\omega$-consistent theory.
Let $L$ be a language containing $L_A = \{+,\cdot,S,0,<\}$, the language of arithmetic. Let $T$ be an $L$-theory.

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*We say $T$ is $\omega$-inconsistent if there is an $L$-formula $\varphi(x)$ in one free variable $x$ such that $T\vdash \varphi(n)$ for every term $n = S(S(\dots S(0)\dots ))$ representing a natural number, but $T\vdash \lnot \forall x\,\varphi(x)$. Otherwise, we say $T$ is $\omega$-consistent.

*$\omega$-logic is first-order logic extended with the infinitary rule with hypotheses $\varphi(0)$, $\varphi(1)$, $\varphi(2)$, $\dots$ and conclusion $\forall x\, \varphi(x)$. We say $T$ is consistent in $\omega$-logic if $T$ does not prove a contradiction in $\omega$-logic.

*We say $T$ has an $\omega$-model if there is a model $M\models T$ such that the underlying set of $M$ is $\omega$, and the symbols of $L_A$ are interpreted in the standard way on $\omega$. In the setting of your question, where $T$ is an $L_A$-theory, the only $\omega$ model is $\omega$ itself (with the symbols of $L_A$ interpreted in the standard way). So $T$ has an $\omega$-model if and only if $\omega\models T$.

Now $T$ has an $\omega$-model if and only if $T$ is consistent in $\omega$-logic, and these conditions imply that $T$ is $\omega$-consistent. But the converse is not true: there are $\omega$-consistent theories $T$ which are not consistent in $\omega$-logic, and hence have no $\omega$-model. An example is given in the section "Arithmetically unsound, $\omega$-consistent theories" on Wikipedia: $\mathsf{PA}+\lnot \omega\mathsf{-ConPA}$, where $\omega\mathsf{-ConPA}$ is the sentence asserting that $\mathsf{PA}$ is $\omega$-consistent.
See also the discussion on this question and this question.
