TITLE: Existence of an $\omega$-nonstandard model of ZFC from compactness
QUESTION [5 upvotes]: 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.

REPLY [6 votes]: I'm not sure, but maybe the question is why a non-ω-model of ZFC must have infinite decreasing $\in$ chains. This follows from two facts:

Viewed externally, any nonstandard model of Peano arithmetic is non-well-founded. Given a nonstandard number $n$, consider the sequence $n-1$, $n-2$, ...
The standard construction of a model of Peano arithmetic within ZFC uses $\in$ for the order relation on the natural numbers. So if the natural numbers within a model of set theory are not well founded, then the model's set inclusion relation is also not well founded. These are, formally, different meanings of "well founded". 

It's interesting to ask how the model can think that its natural numbers are well founded, given that the descending sequence above was completely concrete. The answer is that if that sequence is defined within the model, as $a_k = n-k$, then $k$ can be any number in the model, even a nonstandard one, and the model verifies that this sequence reaches $0$ after $n$ steps.  Only from an external viewpoint can we limit $k$ to an external set of "standard" natural numbers.