TITLE: Why Does The Reflection Principle Fail For Infinitely Many Sentences?
QUESTION [8 upvotes]: I've read the proof that ZF cannot be finitely axiomatized via the reflection principle and the 2nd incompleteness theorem. Since ZF can be countably axiomatized, the finiteness requirement in the reflection principle must be necessary; if the reflection principle worked for infinitely many formulas, the same argument would show that ZF couldn't be countably axiomatized either, as far as I can tell. But I can't figure out why the reflection principle fails for infinitely many formulas.
The reflection principle states that for a finite set of formulas $\{\phi_i\}$ and a starting set $M_0$, we can extend $M_0$ to $M$ so that for each formula $\phi_i$ we have $\phi_i^M (x_1,...,x_n) \iff \phi_i(x_1,...,x_n)$. In Jech, this is proved essentially as follows (thm 12.14): we first take the set of all subformulas of our starting set of formulas $\{\phi_i\}$, which is again finite. Let's say these subformulas are $\{\sigma_1,\sigma_2,\dots\sigma_k\}$ Then we repeatedly "adjoin" elements to our starting set $M_0$ to get a set $M$ so that the following holds for all $u,\dots \in M$: 
$$∃x \ \sigma_j (u, . . . , x) → (∃x ∈ M)\ \sigma_j (u, . . . , x),\ j = 1, ..., k \hspace{1cm} (*)$$
To do this, we begin by letting the $u,\dots$ range over $M_0$, and adjoin an appropriate $x$ value for each formula and each $\sigma_j$ to give $M_1\supseteq M_0$. We can just iterate this procedure a countably infinite number of times to get $M_{i+1}\supseteq M_i$ for each $i$, and let $M$ be the union of the $M_i$. At the end, (*) clearly holds for every $\sigma_j$, because each choice of $u\dots$ will only use variables from some finite iterate $M_i$, and then the desired $x$ for the righthand side lives in $M_{i+1}$. It's easy to conclude from this that $\phi_i^M (x_1,...,x_n) \iff \phi_i(x_1,...,x_n)$, just by building up from the atomic formulas one layer at a time.
I can't find a problem with using this same argument for an infinite set of formulas. We can again just take the set of subformulas, and establish (*) by iteratively constructing $M_{i+1}$ from $M_i$. This should still work because each individual subformula only has finitely many variables. I can't see anything else remotely troublesome for extending the argument. As explained at the start, there must be a problem with this argument, but finding it is eluding me.
Many thanks!

REPLY [7 votes]: The problem is that the formulas and the reflecting models live in completely separate worlds.
The formulas are part of the meta-theory, and the model is part of the universe of the theory. More specifically, while we can internalize finitely many formulas at once (where finite here means in the meta-theory sense), we cannot internalize infinitely many formulas at the same time.
More specifically, since the enumeration of the formulas lives "outside" the world, there is no guarantee that there is a supremum to the sequence of models $M_i$, since the sequence itself might not exist inside the universe.