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3.5 The hierarchy of sets

We define, by transfinite induction, $V_0 = \emptyset $, $V_{\alpha + 1} = P(V_\alpha )$ (power set), and for a limit ordinal $\alpha $,

\[ V_\alpha = \bigcup \nolimits _{\beta < \alpha } V_\beta . \]

Note that each $V_\alpha $ is a transitive set.

Lemma 3.5.1. Every set is an element of $V_\alpha $ for some ordinal $\alpha $.

Proof. See [Lemma 6.3, Jech]. $\square$

In [Chapter III, Kunen] it is explained that this lemma is equivalent to the axiom of foundation. The rank of a set $S$ is the least ordinal $\alpha $ such that $S \in V_\alpha $. By a partial universe we shall mean a suitably large set of the form $V_\alpha $ which will be clear from the context.

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