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  "body": "\n## Definition\n\nA *Grothendieck universe* is a set $U$ such that:\n\n1. If $x \\in U$ and $y \\in x$, then $y \\in U$.\n2. If $x, y \\in U$, then $\\set{x, y} \\in U$.\n3. If $x \\in U$, then the [[power set|00E0YZCERH]] $\\mathcal{P}(x)$ is in $U$.\n4. If $I \\in U$ and $(x_i)_{i \\in I}$ is a family with $x_i \\in U$ for each $i \\in I$, then $\\bigcup_{i \\in I} x_i \\in U$.\n\n## Remarks\n\nThe set $V_\\omega$ of hereditarily finite sets is a Grothendieck universe.\nEach 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.\nThe set $V_{\\omega + \\omega}$ is a non-example: it contains $\\omega$ and each ordinal $\\omega + n$.\nThe union of the family $(\\omega + n)_{n \\in \\omega}$ is $\\omega + \\omega$, which is not in $V_{\\omega + \\omega}$, so clause 4 fails.\nIn ZFC the Grothendieck universes are exactly the empty set, $V_\\omega$, and the sets $V_\\kappa$ for [[inaccessible cardinals|NW4DREQ5DQ]] $\\kappa$.\nMany authors add the clause $\\omega \\in U$, which leaves only the universes $V_\\kappa$ with $\\kappa$ inaccessible.\nThe 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.\n",
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