When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am beginning to have my doubts. We define
\begin{align}
V_0 & :=\emptyset. \\[10pt]
V_{\beta +1} & :={\mathcal {P}}(V_\beta). \\[10pt]
V_\lambda & :=\bigcup_{\beta <\lambda} V_\beta \text{ for any limit ordinal } \lambda,
\end{align}
and finish by saying
$$V:=\bigcup_{\alpha \in \operatorname{Ord}} V_\alpha.$$
However this definition seems to create many problems. I can see at least two immediately: First of all, the Power Set operation is not absolute, that is it varies between models of ZFC. Secondly (and more importantly) this definition seems to be completely circular as we do not know a priori what the ordinals actually are. For instance, if we assume some two, mutually inaccessible large cardinals $\kappa , \kappa'$ to exist, and model ZFC as $V_\kappa, V_{\kappa'}$
respectively, then we get two completely different sets of ordinals! So we seem to be at an impasse:
In order to define the Universe of Sets we must begin with a concept of ordinals, but in order to define the ordinals we need to have a concept of the Universe of Sets!
So my question is to ask: Is this definition circular? The only solution I can think of is that when we define $V$, we implicitly assume a model of ZFC to begin with. Then after constructing the ordinals in this model, we construct $V$ off of them, so to speak. Is this what is being assumed here?
Best Answer
As you noticed, the iterative conception of sets requires a pre-existing universe of sets, and ordinals with which we can label the stages. So if you work within ZFC itself, in other words within an existing model of ZFC, you can perform that iterative construction to obtain $V$. Like Asaf Karagila says here, you cannot get nothing from nothing. Typically, in set theory you work in ZFC, where you have ordinals, and construct $V_k$ for each ordinal $k$. Note that $V$ is not a set, but the entire universe (the one you are working in).
I still think your question is mainly philosophical, your comment to Nik Weaver notwithstanding. After all, you ask:
Of course, since ZFC proves that $V$ is the whole universe, different models of ZFC will have different $V$. If ZFC is consistent, then it has a countable model $V$. That should not be surprising; one can never 'pin down' the set-theoretic universe, not to say using $V$. The same issue shows up with the natural numbers, as Asaf alluded to in a comment; second-order PA with full semantics does not 'pin them down' because our meta-system MS must always be computable, and hence even if MS has (proves existence of) a full semantics model of second-order PA, there are models of MS whose interpretation of the naturals are not isomorphic (if there are any models at all).
We cannot define anything, much less whatever is meant by a "universe of sets", without already working under some assumptions. It is not necessary that you work in ZFC, but what other alternative meta-system do you have in mind? Remember that to construct $V$ you need all those set-theoretic operations that you are using, plus the collection of ordinals, and any meta-system that supports all these is going to look very much like ZF or some extension of it.
Boolos noted the same philosophical circularity in this paper (page 15) (which I rephrased to the language used here and emphasized some points):
To put it more cogently, if you take for granted the power-set operation as a primitive, and start with the empty-set, and also take for granted the ability to consolidate into a single operation the union of any iteratively generated sequence of sets, which may itself be used iteratively to generate more sets, then what you can generate appears to be entirely contained within $V_{δ_1}$. And if you additionally allow taking union of arbitrary (countable) and potentially indescribable sequences, then what you can generate is still contained within $V_{ω_1}$.
The crux is that you cannot generate $V_{ω_1}$ without essentially having $ω_1$. And this corresponds to two logical facts: that there is no countable sequence of ordinals before $ω_1$ whose union is $ω_1$, and that $V_{ω_1}$ is a model of ZF with replacement restricted to countable sequences.
More philosophically, if you envision the stages as being generated rather than pre-existing, then necessarily you cannot generate stage $ω_1$ until you have generated all the stages corresponding to countable ordinals. But there is no way to generate all countable stages without having a generation process that already has length at least $ω_1$. And since $ω_1$ does not appear in any stage up to $V_{ω_1}$, you have no choice but to assume the ability to 'run a generation process' of length $ω_1$ if you want to obtain $V_{ω_1}$ and further stages, which implies that the iterative conception cannot give ontological justification for the existence of $ω_1$.
Just to add, it is true that uncountable well-orderings do appear much earlier than $V_{ω_1}$, but the very fact that $ω_1$ does not appear even at stage $ω_1$ (union of all prior stages) should be a warning that one should not consider all well-orderings of the same length to be on equal footing. In particular, to have a well-ordering as a binary relation on a set that makes it totally ordered with no strictly descending sequence is not the same as being able to iterate along it.
Perhaps someone may find a non-circular way to justify ZFC philosophically, but the iterative conception seems to get us no further than countable replacement.