How should one understand the "universe of sets"?

Question

One way to understand the axioms of $\mathsf{ZFC}$ is to see them as a describing the "universe of sets" $V$, together with the "true membership relation" $\in$. The universe $V$ is built up in stages: \begin{align} V_0 &= \varnothing \\[4pt] V_{\beta+1} &= \mathcal P(V_{\beta})\quad\text{for every ordinal $\beta$,}\\[4pt] V_{\lambda}&=\bigcup_{\beta<\lambda} V_{\beta}\quad\text{for every limit ordinal $\lambda$.} \end{align} with $V$ defined as $\bigcup_{\alpha} V_{\alpha}$.

Although the "universe of sets" as described above is often said to be the intended model of $\mathsf{ZFC}$, this appears to be philosophically contentious. For instance, to make sense of the description of $V$, we need to have a pre-axiomatic understanding of many seemingly non-trivial concepts, such as ordinals.

In addition to this, my understanding is that a model of set theory is typically defined as a pair $(M,\varepsilon)$, where $M$ is a set and $\varepsilon$ is a binary relation on $M$ such that all of the axioms of $\mathsf{ZFC}$ are true when interpreted in $(M,\varepsilon)$. The relation $\varepsilon$ need not be the "true membership relation" $\in$ described earlier, and it appears that in this situation, we study models of set theory in a way that is analogous to how we study models of group theory (i.e. groups). Like most areas of mathematics, model theory seems to take the notion of what a set is for granted.

Since $V$ is "too large" to be a set, it seems that the pair $(V,\in)$ cannot be a model of $\mathsf{ZFC}$ in the formal sense described above. However, even from an informal perspective, the description of $V$ seems unsatisfactory to me: for instance, it seems unclear whether the continuum hypothesis holds in $(V,\in)$, which suggests that the idea of building the universe of sets up in stages is murkier than one might hope.

So, my question is: given the objections raised above, what philosophical positions are commonly held by set-theorists about the "universe of sets"? Where possible, I would be interested in seeing references to the literature.

Answer 1

You may be interested in reading some of Nik Weaver's stuff, maybe Is set theory indispensable? and The concept of a set for starters. If I interpret his position correctly I think he agrees with you that the concept of $V$ is very murky. In particular I think he argues persuasively that the formation of power sets is very murky. Here's a quote from the second paper, emphasis mine:

The iterative conception can also be understood in a different way, not as differentiating sets out of a background universe of collections, but rather as clarifying the concept of a collection as something which must be in some sense “formed” out of elements that in some sense exist “before” it does. Exactly what this means for abstract objects not existing in space and time is hard to pin down, so its status as a clarification is questionable, to say the least. The point is apparently that the existence of a set has something to do with its being constructible, in some obscure sense. What makes this interpretation really incoherent is the fact that each stage of the process by which sets are supposed to be built up involves forming a power set, which in the case of infinite sets is an absolutely non-constructive operation. In other words, the set-theoretic universe is thought of as being built up in an iterative process, yet we pass from each stage of this process to the next in a completely non-constructive way. The power set of the preceding level is not constructed in any sense whatever, it simply appears.