Von Neumann Universe V, rank and every set contained in V

Question

I sought a consistent definition for the rank of set and I found the definition of Von Neumann universe to define the rank of a set as follows:

The cumulative hierarchy is defined by transfinite recursion over all ordinals by

• $V_0=\emptyset$,
• $V_{\alpha+1}=\mathcal P(V_\alpha)$, and
• $V_\alpha=\bigcup_{\beta<\alpha}V_\beta$

Then, the rank of a set $S$ is the smallest $\alpha$ such that $S \subseteq \alpha$. But I don't understand how does any mathematical set (for instance a real set) could be include in a $V_\alpha$ whereas $V_\alpha$ contains a construction of empty set like that

    V0 = { },
    V1 = {{ }},
    V2 = {{ }, {{ }}},
    V3 = {{}, {{ }}, {{ }, {{ }}}},
    etc.

I saw that Von Neumann set the natural set with that method, but in the ZFC theory is there a result that proves it exists a bijection between any set and that ranked construction of Von Neumann please?

I have never studied neither the set theory nor the ZFC theory before but I would like to understand that point.

Thank you in advance for your answer.

Answer 1

First of all, your $V_3$ is wrong - it should have four elements. $V_3=\{\emptyset, \{\emptyset\}, \{\{\emptyset\}\}, \{\emptyset,\{\emptyset\}\}\}$. (I've abbreviated "$\{\}$" as "$\emptyset$" for readability.)

A good exercise: what's the cardinality of $V_n$?

You ask where in $V$ do common mathematical objects, like $\{1\}$, lie. Well, the view of set theory is that mathematical objects are sets built up from the empty set. E.g. $0$ is $\{\}$, $1$ is $\{\{\}\}$, etc. - these are the finite ordinals. A rational number is an equivalence class of ordered triples of finite ordinals (sign, numerator, denominator), and a real number is a set of rational numbers. (This is merely one way to do the coding; there are other ways.) The role of set theory as a foundation of mathematics comes from the observation that all of mathematics can be developed within $V$. It's difficult to make this precise, since what is "all of mathematics?", but there is a lot of writing on this subject.

There are also plenty of objections to this approach, on the grounds that it is extremely artificial and nonintuitive; see e.g. this MSE question addressing the treatment of ordered pairs specifically, and this mathoverflow question addressing some of the artificial phenomena which arise in general from this approach. A very different sort of object, structural set theory, is often advocated in this context. One intermediate approach is via set theory with urelements, and this has the benefit of not being too different from classical set theory.