The Cumulative Hierarchy

Where do all the sets actually live? The ZFC axioms tell you which sets may be built, but they don't draw the map. The cumulative hierarchy is that map: a single towering structure, built in stages indexed by the ordinals, that contains every set exactly once. Start from nothing, and at each stage form the power set of what you have so far. Written V_\alpha for the sets available by stage \alpha:

Base: V_0 = \varnothing — start with nothing.
Successor: V_{\alpha+1} = \mathcal{P}(V_\alpha) — each stage is all subsets of the previous one.
Limit: V_\lambda = \bigcup_{\beta < \lambda} V_\beta — gather everything from all earlier stages.
The whole universe is V = \bigcup_{\alpha \in \mathrm{Ord}} V_\alpha — every set appears in some V_\alpha.

It is exactly the shape of a transfinite recursion: a base, a successor rule, and a limit rule — the three cases again.

The universe as a widening cone

Picture V as an upward-widening cone. The pointed bottom is the empty stage; each layer above is strictly wider — it holds every subset of the layer below — and the cone keeps opening out through the finite stages, past the first limit stage V_\omega, and up through every ordinal without end. Climb it stage by stage:

The finite stages fill up fast: V_0 = \varnothing, V_1 = \{\varnothing\}, V_2 = \{\varnothing, \{\varnothing\}\}, and in general |V_{n+1}| = 2^{|V_n|} — the sizes rocket 0, 1, 2, 4, 16, 65536, \dots. The first infinite stage V_\omega collects all the finite stages: it is precisely the set of all hereditarily finite sets — finite sets whose members are finite sets whose members are… all the way down.

// Sizes of the finite stages: |V_0| = 0, and |V_{n+1}| = 2^|V_n|. let size = 0; // |V_0| console.log("|V_0| = " + size); for (let n = 0; n < 5; n++) { size = Math.pow(2, size); // |V_{n+1}| = 2^|V_n| console.log("|V_" + (n + 1) + "| = " + size); } console.log("V_omega then unites them all: the hereditarily finite sets.");

Rank: which stage a set first appears

Because every set turns up somewhere, each set x has a rank: the least ordinal \alpha with x \in V_{\alpha+1} — the stage where it is first born. Rank is defined by recursion, \operatorname{rank}(x) = \sup\{\operatorname{rank}(y) + 1 : y \in x\}, and it is the ruler that measures how "deep" a set is. The claim that every set has a rank — that V = \bigcup_\alpha V_\alpha catches everything — is equivalent to the Axiom of Foundation. Foundation is exactly what forbids the pathological loops (a set containing itself) that would have no birthday in the hierarchy.

Because nothing is ever thrown away — each stage contains all the earlier ones. Since V_\alpha \subseteq V_{\alpha+1} = \mathcal{P}(V_\alpha) (every element of V_\alpha is also a subset of it, hence a member of its power set), the stages are nested: V_0 \subseteq V_1 \subseteq V_2 \subseteq \cdots. The universe accumulates. This nesting is what lets the cone widen monotonically — a set that appears at stage 5 is still sitting there at stage 500 and at every stage after.

$V$ is not a set — it's a proper class. If V were a set it would live in some V_\alpha, hence contain itself — impossible. The stages V_\alpha are all sets; their union over all ordinals is a proper class, too big to be a set.
$V_\omega$ is not all sets — only the hereditarily finite ones. The reals, and \omega itself, first appear at V_{\omega+1} and beyond. The interesting infinities live above the first limit stage, not at it.