TITLE: What is wrong with this "proof" that there is no $\omega$th inaccessible cardinal?
QUESTION [5 upvotes]: "Theorem": There is no $\omega$th inaccessible cardinal.
"Proof":
Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, $V_{\kappa_n}$ is a model of ZFC + "there are $n-1$ inaccessible cardinals."
Now suppose there are $\aleph_0$ many inaccessible cardinals. Then in particular we can take the $\omega$-th inaccessible $\kappa_\omega$, and there will still be $\aleph_0$ many inaccessibles below it. Thus $V_{\kappa_\omega}$ is a model of ZFC+"there are $\aleph_0$ many inaccessible cardinals", and so by the (second) incompleteness theorem that theory is inconsistent. $\square$
But this "result" is literally too good to be true. So where am I wrong here?

Edit: Yep, I realize now what the fallacy is. Daniel Fischer and GME are correct - the existence of $\aleph_0$ many inaccessibles does not imply the existence of an $\omega$th inaccessible, just as the fact that there are $\aleph_0$ many natural numbers does not imply the existence of the ordinal $\omega$. 
In other words, there can be models of ZFC + "there are $\aleph_0$ many inaccessibles" such that every inaccessible has finitely many inaccessibles below it, so the argument fails. In fact $V_{\kappa_\omega}$ is just such a model: it knows of $\kappa_n$ for all finite $n$, but it sees $\kappa_\omega$ as a proper class.

REPLY [3 votes]: I don't see the problem with your proof, but I look at it like this. Let $\theta_0$ be the first (I assume always strongly) inaccessible cardinal. Let $\theta_1$ be the smallest inaccessible larger than $\theta_0$, $\theta_2$ be the smallest inaccessible cardinal larger than $\theta_1$ and so forth. So what is $\theta_\omega$? Method 1 says it is the sup or limit of all of the $\theta_i$ as $i$ ranges over the finite ordinals. Method 2 says it is the smallest inaccessible cardinal larger than all the $\theta_i$. These two are NOT the same. The cardinal defined by Method 1 is the limit of a countable set, namely $\theta_i$, so it is accessible; in fact it is singular. The next inaccessible cardinal is therefore $\theta_{\omega+1}$. So it appears that a skip appears every time we hit a subscript that is a limit ordinal; in particular, the set of all inaccessibles less than a large cardinal $\kappa$ is not a club set.