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I would like to know how to obtain not well-founded $\omega$-models of ZFC.

Are there any books about it? Other references to the literature are also welcome.

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    $\begingroup$ I don't understand the votes to close - this is a reasonable thing to wonder about, and it is hard to find resources on it. $\endgroup$ – Noah Schweber Jan 31 '17 at 22:29
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    $\begingroup$ Additionally it's bizarre that this has been closed as "unclear" - it's perfectly clear. I wonder how many of those who voted to close are actually familiar with the topic . . . $\endgroup$ – Noah Schweber Mar 7 '17 at 18:16
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Once you know that the consistency of "There is a transitive model of ZFC" implies the consistency of "There is an $\omega$-model of ZFC", assuming the existence of the former will provide you with the existence of the latter.

Of course, it is consistent that there are no $\omega$-models, even if ZFC is consistent. So just obtaining them out of a model of ZFC is impossible.

But here is a nice way to get what you want. Start with a transitive model of ZFC, $M$. Fix some regular $\kappa>\omega$, and now add a generic ultrafilter and consider the generic ultrapower of $M$. It will have critical point $\kappa$, so it remains an $\omega$-model, but it will be ill-founded, as long as you didn't use a precipitous ideal for your forcing.

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    $\begingroup$ +1. Another (unrelated of course) way to get an illfounded $\omega$-model is via a Barwise Compactness argument: this lets us get an $\omega$-model with wellfounded part of height $\omega_1^{CK}$, or indeed of height $\alpha$ for any admissible ordinal $\alpha$. $\endgroup$ – Noah Schweber Jan 31 '17 at 22:28

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