I have read several times that assuming Con(ZFC), and using compactness it can be proved the existence of a model of ZFC with an ill-founded $\omega$. How is that? Any reference will be welcome.



Existence of an $\omega$-nonstandard model of ZFC from compactness
This is a standard application of the Compactness Theorem, and works basically the same in producing nonstandard models of ZFC as it does for producing nonstandard models of PA or real-closed fields.


Consider the theory $T$, in the language of set theory
augmented with an additional constant symbol $c$,
consisting of all the ZFC axioms, plus the assertions that
$c$ is a natural number, but not equal to $0$, not equal to
$1$, and so on, including for each natural number $n$ the statement $\varphi\_n$ that $c$ is not
equal to $n$. (Note that 
all such $n$ are definable in set theory, and so we use the
definition of $n$ in $\varphi\_n$ when asserting that $c$ is
not $n$.)


If there is a model $M$ of ZFC, then every finite subtheory
of $T$ is consistent, since any finite subtheory of $T$
makes only finitely many assertions about $c$, and we may
therefore interpret $c$ as any natural number of $M$ not
mentioned in the subtheory.


Thus, by compactness, $T$ has a model. Any such model of
$T$ will be $\omega$-nonstandard, since the interpretation
of $c$ in the model will be a nonstandard natural number.


There are numerous other ways to produce
$\omega$-nonstandard models of ZFC from existing models,
the most common being ultrapowers.


The conclusion is that if there is any model of ZFC, then
there are nonstandard models of ZFC. The converse of this
is not true, for if there are any transitive models of ZFC,
then there is an $\in$-minimal such model $M$, and being
standard, $M$ will have the same arithmetic as the ambient
universe, thus thinking Con(ZFC), but being minimal, will
have no transitive models of ZFC inside it.


