Reflection theorem for all of ZFC true "Platonistically" I am reading Kunen's "Set Theory: An Introduction to Independence Proofs," and I am getting confused about the reflection theorems (Section IV.7).
Given any finite list of ZFC axioms, the reflection theorems prove that there is a $V_\alpha$ in which the formulas are absolute (Theorem 7.5), then that there is a countable model where they are absolute (Theorem 7.8), and finally that there is a countable transitive model that satisfies the formulas (Theorem 7.10)
But then he says that "Platonistically" the arguments could be carried out on the entire list of ZFC axioms.   I am confused about several things.  First, what does it means to prove something Platonistically?  The theorems, as they stand are, I believe, proofs within ZFC.   One might also do a proof in the metatheory, establishing that it is true that there is a set in the ZFC universe with some property, but without there being a proof of this within the ZFC universe.   It seems that if we were doing a metatheory analog of Theorem 7.5, it would be easy to extend the proof to handle all the axioms, and not just a finite subset.  Is this what it means it to be Platonistic?   So what is wrong with this, and why is it disparaged as Platonistic (and I guess, considered irrelevant and not useful)?
My second question is why this does not cause some problem with an infinite regress (along the lines of his Theorem 7.7), where each ZFC model contains a smaller ZFC model?
I am new to this subject, so I apologize if my question seems poorly explained.   I feel like I cannot progress to the next chapter until I have a better picture of this. 
 A: Arguing "platonistically" means assuming there really is a universe of sets. A (set) model of ZFC is an approximation to it, but far from the real deal. So, there is no infinite regress problem because even if the true universe of sets has many stages $V_\alpha$ that are elementary substructures, these structures do not need to have such stages themselves, and there is actually not much of a reason to expect that they do. This is because even if we believe that the universe has this strong reflection property, this would be a true property of $V$, but not a theorem of ZFC, or even a consequence of ZFC true in all transitive structures, or anything of the kind. And if you just start with an arbitrary model of ZFC (or even an arbitrary transitive model), there is no reason to believe it will capture this particular property of the universe. (We lose something through the reflection process, so to speak.)
But if you are not comfortable thinking in these terms, the above can be somewhat confusing. The thing is this: If $M$ is a structure in a countable language, the Löwenheim-Skolem theorem ensures that there is a countable elementary substructure $N\preceq M$. The issue here is that if you do not start with a set structure but with the actual universe of sets, and attempt to formalize the proof of the Löwenheim-Skolem theorem you run into subtle difficulties (related, for instance, to Tarski's undefinability of truth). The reflection theorem is the formal surrogate we get instead. But "morally" it feels as if the full Löwenheim-Skolem theorem should just hold (and note that this would give us something stronger than just reflection of the entire list of ZFC axioms), and it is only a (bizarre?) insistence in working within the confines of first-order logic that prevents us from having it.
In any case, the fact that we cannot straightforwardly formalize this desired strengthened version of reflection is no accident. Simply, if there is some stage $V_\alpha$ that models ZFC, then there is a first one, and that stage cannot reflect to a smaller $V_\beta$; this is the regress argument. But even if we do not insist that the reflection be to a stage $V_\gamma$, there are problems. For instance, say we start with an uncountable model $M$ of ZFC+"ZFC is inconsistent". By the Löwenheim-Skolem theorem (say, in the metatheory, if you will), it has countable elementary substructures. But they better not be elements of $M$, since they would witness, within $M$, the consistency of ZFC, which is impossible by design. This means that we cannot really work around the issue and prove a "strengthened" reflection theorem. 
Be that as it may, the above is truly a powerful heuristic, and it is not uncommon to hear informal presentations of an argument that start by considering a countable elementary substructure of the universe, or some such. In reality, what one does is to start with a $V_\alpha$ that reflects "enough" of the universe, and then take a countable elementary substructure of it. Another common approach is to work not with the entire universe but with some $H_\kappa$, and then take small elementary substructures which (by definition) belong to $H_\kappa$. This allows us to establish combinatorial properties of the universe by arguing "locally" about small models (as if the informal "platonistic" approach were true after all). 
(There are additionally subtleties. For instance, there is a formula that defines -- provably in ZFC -- a countable structure $(M,E)$, and for each ZFC axiom $\phi$, ZFC proves that $(M,E)$ is a model of $\phi$. But this does not mean that ZFC proves that $(M,E)$ is a model of ZFC.) 
