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According to Cantor's Attic (link):

Not all transitive models of ZFC have the $V_\kappa$ form, for if there is any transitive model of ZFC, then by the Löwenheim-Skolem theorem there is a countable such model, and these never have the form $V_\kappa$.

Question: what do countable transitive models of ZFC look like?

It is an interesting fact that every transitive model of second-order ZFC equals $V_\kappa$ for some $\kappa$. See Asaf's answer here.

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goblin GONE
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    The second question was asked, by you as well I believe, on this site before. It is not just $V_\kappa$, but $\kappa$ is inaccessible. – Asaf Karagila Jul 07 '13 at 10:36
  • @AsafKaragila, do you mean this question? You're essentially correct - since your answer states the answer to my second question. – goblin GONE Jul 07 '13 at 10:39
  • Also http://math.stackexchange.com/questions/307901/does-tg-prove-that-zfc2-has-a-model/307905#307905 – Asaf Karagila Jul 07 '13 at 10:43
  • @AsafKaragila, I edited the question linking to an answer to the second part. Sorry about 'missing' these sorts of things first time around, I often have to reread an answer many months later before things truly sink in. – goblin GONE Jul 07 '13 at 10:48
  • Yes, that's how it usually goes. sometimes years, too. You might also want to edit the title here. – Asaf Karagila Jul 07 '13 at 10:49
  • I've often wondered this myself. In particular, I've wondered about visual representations of such things. For instance, if we were to draw the usual sorts of inner model pictures that we do for, e.g $L$, what would we draw for some countable transitive model. In particular, where do you label objects like $\aleph_1$ (not $\aleph_1^M$)? Does that `cap' the model in the same way that $\kappa$ does for $\kappa$ inaccessible for the smallest model of $ZFC_2$? – Neil Barton May 22 '14 at 22:05
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    @Neil: Every member of a countable transitive model is a countable set. So $\omega_1$ is not in any such model. So you just don't draw it in your diagram, or you draw it outside. And moreover, since there are only countably many ordinals in a countable model, almost all the countable ordinals are outside the countable model too. In fact, it is consistent that all the transitive models have the same ordinals, some countable ordinal $\alpha$. So the comparison with an inaccessible cardinal doesn't hold very well here. – Asaf Karagila May 23 '14 at 01:02
  • Just to be clear I was (obviously) not suggesting that $\omega_1$, is in the model, but rather that it might be the "cap" in the same way an inaccessible is the "cap" in the smallest model of $ZFC_2$ but not in the model. What you say about almost all ctble ordinals not being in a ctble model shows why this analogy is bad; I take it that the argument is that the set of all ctble ordinals is unctble, any ctble transitive model contains only ctbly many ordinals, and so there are unctbly many ctble ordinals not in the model, and thus the model does not contain `almost all' ctble ordinals. Thanks! – Neil Barton May 23 '14 at 14:00

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That's a tough question to answer. If $M$ is a countable transitive model of $\sf ZFC$, and $M$ is a model of $V=L$ then $M=L_\beta$ for some countable $\beta$. But other than similar cases like that, it's very hard to say exactly how it looks like.

To illustrate the point, if we have some very large cardinals in the universe then we can take an elementary submodel of some $V_\kappa$ which contains a lot of large cardinal assumptions. The countable model will think that a lot of countable ordinals are very large cardinals, which makes the model quite large and complicated, but when considering a countable model of the same theory it's difficult to explain how it looks like.

Also over countable models we can prove that generic sets exist, therefore we can force over them and generate new countable transitive models which are very different. So we can force and add anything that can be added by forcing, or class forcing.

All in all we can say these things:

  1. If $M$ is a countable transitive model of $\sf ZFC$ then ${\sf Ord}^M=\beta$ for some countable ordinal $\beta$, and $L_\beta$ is a countable transitive model of $\sf ZFC+\it V=L$.
  2. Every model of $\sf ZFC$, and even more so when the model is transitive, is the limit of its own von Neumann hierarchy.
Asaf Karagila
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