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  "body": "\n## Definition\n\n*Von Neumann–Bernays–Gödel set theory*, written NBG, is the first-order theory in the language $\\set{\\in}$ whose variables range over [[classes|W473FH7Z0P]].\nA class $X$ is a *set* if $X \\in Y$ for some class $Y$, and a *proper class* otherwise.\nThe axioms of NBG are these.\n\n1. Extensionality: classes with the same elements are equal, as in the [[axiom of extensionality|0TB16T2C97]].\n2. Pairing, union, power set and infinity: as in [[ZF|0HE5WDQ9FJ]], stated for sets.\n3. Foundation: the [[axiom of regularity|6QXBY7J0M5]], for all nonempty classes.\n4. Replacement: if $F$ is a class that is a function and $a$ is a set, then the image $F[a]$ is a set.\n5. Class comprehension: for each formula $\\varphi(x)$ in which all quantifiers range over sets, possibly with class parameters, there is a class $\\set{x : x \\text{ is a set and } \\varphi(x)}$.\n6. Global choice: there is a class function that picks an element from every nonempty set.\n\n## Remarks\n\nClass comprehension can be replaced by finitely many of its instances, so NBG has a finite list of axioms; ZF has no finite axiomatization if it is consistent.\nNBG is a conservative extension of ZFC: a statement about sets is provable in NBG if and only if it is provable in ZFC.\nExample: the class $V = \\set{x : x = x}$ of all sets exists by comprehension, with $\\varphi(x)$ the formula $x = x$.\nNon-example: $V$ is not a set.\nIf it were, then comprehension would give the class $R = \\set{x : x \\notin x}$, which would be a set as a subclass of $V$, and then $R \\in R \\leftrightarrow R \\notin R$.\nMorse–Kelley set theory is the stronger theory in which the formula in class comprehension may also quantify over classes.\nSome authors drop global choice, or replace it by the [[axiom of choice|0HTGB13FE2]] for sets; NBG without global choice is conservative over ZF.\n",
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