TITLE: Is V, the Universe of Sets, a fixed object?
QUESTION [36 upvotes]: 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?

REPLY [2 votes]: I come a little late, just to clearly state and sum up something that was began to be outlined by some here.
Namely, the syntacticalist view I have that nothing in math or formal sciences in general is a "fixed object" unless its's an actual syntactic object; otherwise it's just a concept in one(s)'s mind(s), possibly formalized into actual syntax and/or conceived from actual syntax. And the concepts themselves are subjectively as diverse as conceivers, only given a same name if those agree enough about their intuitions.
So here, the concepts of π, the Klein Four-Group or the Von Neumann Universe aren't more fixed than by traditional common shared intuition.
But, where the question becomes interesting (to me) and your answer quite true (as your intuition had guessed), is once rephrased given all the above. Sort of: "Has the concept of Von Neumann Universe any concrete syntactical counterpart, an actual avatar, such as π or the Klein Four-Group?"
And the answer indeed is: not really. Yes π has any algorithmic recursive enumeration of its digits in any fixed base. Yes Klein's Group has his Cayley Table.
Though V only has, as said Nate Eldredge in a comment, its defining predicate - as a unary predicate symbol for e.g. ZFC, as a constant symbol of class for e.g. NB. However as this phrase with a hyphen shows, as a defining Formula it really is itself well-"defined" only relatively to some fixed formal system, for which it actually defines something local. Finally, it doesn't alone have an actual independent existence that could unequivocally manifest all its features; only as a string of symbols devoid of most of the answers to the questions one can ask about V as a concept, not at all intrinsically tied to this intuition of ours, enough so to embody it into an actual reality.