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After reading books before lectures, here's my thought regarding inner models and so on. Correct me if I am wrong.

So there's universe $V$, which we assume to be the true universe. By Gödel's incompleteness theorem (and the need to assume large cardinals etc.), we cannot prove that inner models that are subclass of the universe exist, but if they do, then the inner model satisfies the ZF(C) theory (axioms). Standard model is an inner model that uses actual element relation $\in$ in $V$ - we cannot prove the existence of it again by Gödel's incompleteness theorem. Sometimes inner models refer to standard models. Standard models may be both internally and externally well-founded but may not be transitive - in such case, Mostowski collapse lemma guarantees that there exists a unique transitive class model. If the model is not well-founded externally, then $V$ would see it as not well-founded so the talks about Mostowksi collapse lemma are of no use. However, as $V$ is assumed, we may instead assume that this model is universe $V$.

If standard model exists, there is a minimal countable model that internally satisfies ZF(C) theory, but this model is externally not well-founded, while standard model is.

Is this correct understanding?

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Not quite. We can always define $L$, for example, and we can prove that for each axiom of ZFC, its relativization to $L$ is true.

Similarly we can prove that such relativization holds for other definable classes. Although they may or may not be equal to $L$, depending on some properties of $V$. The incompleteness theorem tells us that we cannot prove the consistency of ZFC, or rather an effective subtheory of the axioms of the universe (ZFC and some more). So we can't truly define a truth predicate and claim that $L$ satisfies all the axioms of ZFC. We can just check them one by one.

Finally, standard models are set models, whereas inner models are usually proper classes, but other than that your understanding is correct. Well-founded models are indeed those that are well-founded externally, i.e. in $V$.

And if any model exists in $V$, then by internal well-foundedness of $V$ there is a minimal model which cannot be well-founded. The reason is that well-founded models agree with $V$ about basic number theoretical statements such as Con(ZFC). So if there is a model in the universe, $V$ thinks that the integers satisfy Con(ZFC). However a minimal model doesn't include a model, so it cannot agree on Con(ZFC), and the only way that happens is if this model has non-standard integers, so it cannot be well-founded.

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    $\begingroup$ And I wrote all that on my phone. Phew.. $\endgroup$ – Asaf Karagila Mar 25 '14 at 1:16

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