Can a theory be consistent but not $\omega$ consistent? Say we have an axiomatizable theory $T$ extending $Th(A_E)$ where $A_E$ are the axioms of arithmetic. Is it possible to extend $T$ such that our extended theory is consistent but not $\omega$-consistent?
Recall that a theory $T$ is defined to be $\omega$-consistent if for each formula $\psi$, if $T \vdash \neg \psi(a)$ for each $a \in \mathbb{N}$, then $T \nvdash \exists b \psi(b)$.
Here are my thoughts: We need to add a formulas to $T$ such that for some $c$, we have $T \vdash \psi(c)$ and for all $a$, $T \vdash \neg \psi(a)$. I am not sure how it is possible to prove these two statemetns while maintaining that $T$ is consistent since we would have that we could prove $\psi(c)$ for some constant $c$ and $\neg \psi(c)$ since $c$ is in our domain. What am I missing here?
 A: PA + "PA is inconsistent" is indeed a consistent theory - this is the conclusion of Gödel's incompleteness theorem . 
But there are simpler examples. For instance, start with PA and add a new constant $x$ and an infinite sequence of axioms of the form "$x > 1$", "$x > 2$", "$x > 3$", ... -  this theory is consistent by the compactness theorem. It proves $(\exists y) [y = x]$, but it also proves $x \ne n$ for every numeral $n$, because of the extra axioms.
A: Whenever a set theory has surprisingly low consistency strength because of the compactness theorem, it tends to be $\omega$-inconsistent with the non-existence of the large cardinals that you would expect the theory to imply.
For example, let $T$ be the theory $\text{ZFC + "There is a correct cardinal}"$ where a cardinal $\kappa$ is correct if $V_\kappa \prec V$ and we formalize this with a constant $\kappa$ in the language of $T$ and a schema saying that $V_\kappa \prec V$. Since $T$ is conservative over ZFC, $T+ \text{"There is no worldly cardinal"}$ is eqiconsistent with ZFC (and even arithmetically sound if ZFC is) but $\omega$-inconsistent because (in any model of this theory) the non-worldliness of $\kappa$ means that the model thinks that some axiom of ZFC fails in $V_\kappa$ but every standard axiom holds in $V_\kappa$ since $V_\kappa \prec V$. Thus $T$ is $\omega$-inconsistent with any axiom implying that the class of wordly cardinals is non-stationary, where stationarity is defined with respect to the definable classes, or equivalently the classes of $\Sigma_n$-correct cardinals for every $n$.
Similarly, Feferman theory, the theory saying that the correct cardinals are unbounded (formalized with a predicate $C$, an axiom saying that the class $C$ is unbounded and a schema saying that every element of $C$ is a correct cardinal) is equiconsistent with ZFC but every extention saying that there are not otherwordly chains of every finite length is $\omega$-inconsistent. If we take Feferman theory to include the axiom of replacement for formulas involving $C$, any extention  saying that there are not otherworldly chains of every ordinal length is $\omega$-inconsistent.
The Levy scheme, which says that there is a correct cardinal which is also inaccessible, is equiconsistent with Ord is Mahlo, the theory saying that there is an inaccessible cardinal in every definable closed unbounded class, but Levy scheme + "There is no $\Sigma_\omega$-Mahlo cardinal" is $\omega$-inconsistent because if $\kappa$ is $\Sigma_n$-correct and inaccessible, the inaccessible cardinals are $\Sigma_n$-stationary in $V_\kappa$, so if $\kappa$ is fully correct and inaccessible, $\kappa$ is definably stationary with respect to the metatheory but the non-$\Sigma_\omega$-Mahloness of $\kappa$ means that the model thinks that there is a subset of $\kappa$, definable over $V_\kappa$, which contains no inaccessible cardinal.
$WA_0$, the weakest version of the wholeness axioms is equiconsistent with ZFC+ the schema saying that for every metatheoretical natural number $n$ there is an $n$-huge cardinal, as pointed out in this MathOverflow answer by Dmytro Taranovsky, and thus weaker than the existence of a $\lt \omega$-huge cardinal (by which I mean one that is $n$-huge for every $n$, quantifying internally in the model) but $WA_0$+"The critical point of the wholeness embedding is not $\lt \omega$-huge" is omega-inconsistent since the critical point is $n$-huge for every standard natural $n$.
