# What do countable transitive models of ZFC look like?

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.

• 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 7 '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 Jul 7 '13 at 10:39
• – Asaf Karagila Jul 7 '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 Jul 7 '13 at 10:48
• @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 1:02

## 1 Answer

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.