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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.

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    $\begingroup$ Maybe useful: The Philosophy of Set Theory. Also Penelope Maddy's publications. $\endgroup$
    – Mauro ALLEGRANZA
    Sep 27 at 13:31
  • $\begingroup$ You are welcome to intuitively let V = {x | x = x} be the universe, if that feels more natural. One benefit of building the universe by stages, is that we create The Cumulative Heirachy, where $V_α$ can serve as models of ZF for various α. In conjunction with the reflection theorems, truth in one level can tell us about truth in other levels, and in general we can go on to construct many models using levels in the Cumulative Heirarchy. Note: Your notion of model is a bit restrictive, you don't need that M is a set, just that M is a class. $\endgroup$
    – Michael Carey
    Sep 27 at 16:14
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    $\begingroup$ As to the philosphical point: Kunen, in his "An Introduction to Independence Proofs" says to think of the Axioms, not as describing the universe of sets, but rather a restriction of the domain of discourse to something appropriate for mathematics-as well as the closure properties of such a restriction. There is an important addition to your notion of model. M is model of ZFC if and only if, assuming ZFC we prove that ZFC holds in M. In the case of V, if we assume ZFC- then it is hopefully obvious that Choice holds in V $\endgroup$
    – Michael Carey
    Sep 27 at 16:31
  • $\begingroup$ Your intuitive doubt is sensible. In fact, the iterative conception fails to justify full replacement in ZFC, with an explicit citation of Boolos. Ping me if you want any clarification. $\endgroup$
    – user21820
    Sep 30 at 2:55
  • $\begingroup$ @user21820: Hello. Thank you for the link. I would certainly like to discuss this further in chat. If you are available, then please let me know. I'm just about to eat breakfast, but I should be free in 30 minutes if that works for you :) $\endgroup$
    – Joe
    Oct 1 at 9:14

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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.

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  • $\begingroup$ To add technicalities, for anyone interested.The part about the Cumulative Heirarchy formulating sets, as being "constructed" from elements which exist before them is referencing this fact: Define: Rank(x) = least α s.t. x ⊆ $V_α$. Thus by definition x∈$V_δ$, where δ = α + 1 (RAA) notice that if x ∈$V_α$ ( or any lower level), then it is a subset of some βth level < α, contradiction. thus x is truly born on the α+1th level, and its "parents/elements" are contained in earlier levels. $\endgroup$
    – Michael Carey
    Sep 27 at 22:10
  • $\begingroup$ further, we can now formally say φ(α,x) = "x is a set" means rank(x) = α, i.e. we can talk about the existence of sets in terms of definable formula- this is what the post above likely meant by "existence of a set has something to do with it being constructable" ( sometimes the word definable is used instead) this simple way of refering to sets is the begining of the work that lead Cantor's contribution to showing that CH is independent of ZFC. ( Cantor got half way there, Cohen did the other half) $\endgroup$
    – Michael Carey
    Sep 27 at 22:18
  • $\begingroup$ Thanks for this answer. While I am not sure whether I buy all of Nik Weaver's comments, I think he offers a reasoned and intelligent philosophical position. It is also reassuring to know that much of mathematics can be proved in much weaker axiom systems, and with much more modest philosophical commitments about the nature of sets. $\endgroup$
    – Joe
    Sep 29 at 18:00

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