The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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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