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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.

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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.

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  • $\begingroup$ Thank you for your expanded answer but you haven't completely answered my question . You say "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", I already understood that when I talked about the construction of natural set with the Von Neumann method, but is there a result that proves it exists a kind of "surjection" between the construction from empty set and any set? May we exhibit a case where we don't construct the set of any possible sets with VN method but where this result stay true? Thank you in advance. $\endgroup$ – Moth Mar 8 '17 at 16:34
  • $\begingroup$ @Wmog I'm not sure I understand your question, but maybe you're asking whether we can prove in ZFC that every set is in $V$? The answer is yes to this question, and the crucial axiom is Foundation: if $\alpha_0$ were a set not in $V$, it would have to have an element $\alpha_1$ not in $V$; and proceeding this way we can construct a sequence of sets $\alpha_0\ni \alpha_1\ni \alpha_2\ni ...$, contradicting Foundation. Does this answer your question? If this is not your question, can you elaborate on it? $\endgroup$ – Noah Schweber Mar 8 '17 at 19:17

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