Grothendieck universe

Remarks

The set $V_\omega$ of hereditarily finite sets is a Grothendieck universe. Each clause holds because elements, pairs, power sets and finite unions of hereditarily finite sets are again hereditarily finite, and an index set $I \in V_\omega$ is finite. The set $V_{\omega + \omega}$ is a non-example: it contains $\omega$ and each ordinal $\omega + n$. The union of the family $(\omega + n)_{n \in \omega}$ is $\omega + \omega$, which is not in $V_{\omega + \omega}$, so clause 4 fails. In ZFC the Grothendieck universes are exactly the empty set, $V_\omega$, and the sets $V_\kappa$ for inaccessible cardinals $\kappa$. Many authors add the clause $\omega \in U$, which leaves only the universes $V_\kappa$ with $\kappa$ inaccessible. The axiom of universes states that every set is an element of some Grothendieck universe; it is equivalent to the existence of arbitrarily large inaccessible cardinals.