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  "body": "\n## Definition\n\nWork in [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].\nLet $\\varphi(x, p_1, \\dots, p_n)$ be a formula of [[first-order logic|086QKPF0A6]] in the language $\\set{\\in}$, and let $p_1, \\dots, p_n$ be sets, the parameters.\nThe *class* defined by $\\varphi$ with these parameters is the collection\n$$C = \\set{x : \\varphi(x, p_1, \\dots, p_n)},$$\nand $x \\in C$ is an abbreviation for $\\varphi(x, p_1, \\dots, p_n)$.\nA class $C$ is a *set* if there is a [[set|0SJ3RX9TXA]] $a$ with $\\forall x\\, (x \\in a \\leftrightarrow x \\in C)$.\nOtherwise $C$ is a *proper class*.\n\nIn class theories such as von Neumann–Bernays–Gödel set theory, classes are instead objects of the theory.\nThere, a set is a class that is an element of some class.\n\n## Remarks\n\nExample: the class $\\set{x : x \\subseteq \\omega}$ is a set, namely the power set of $\\omega$.\nNon-example: the class $V = \\set{x : x = x}$ of all sets is not a set, so it is a proper class.\nIf $V$ were a set $a$, then separation would give the set $\\set{x \\in a : x \\notin x}$, and [[Russell's paradox|0HESMMSYEG]] would follow.\nThe class of all [[ordinals|0RDJK7EQYY]] is also a proper class, by the Burali-Forti argument.\nEvery set $a$ is a class, defined by the formula $x \\in p$ with parameter $p = a$.\n",
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