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  "body": "\n## Definition\n\n*Naive set theory*, in its formal reading, is the [[set theory|JRBRNZT680]] in first-order logic with the relation symbol $\\in$ and these two axioms.\n\n1. Extensionality: two [[sets|0SJ3RX9TXA]] with the same elements are equal, as in the [[axiom of extensionality|0TB16T2C97]].\n2. Unrestricted comprehension: for each formula $\\varphi(x)$, possibly with parameters, in which $y$ is not free, $\\exists y\\, \\forall x\\, (x \\in y \\leftrightarrow \\varphi(x))$.\n\nIn a wider sense, naive set theory is the informal use of sets in ordinary language, without a fixed list of axioms.\n\n## Remarks\n\nExample: the comprehension instance for $\\varphi(x) = (x \\ne x)$ gives a set with no elements, the empty set, and extensionality makes it unique.\nNon-example: the formal theory is inconsistent, so it has no model.\nIndeed, take $\\varphi(x) = (x \\notin x)$; comprehension gives $y$ with $\\forall x\\, (x \\in y \\leftrightarrow x \\notin x)$, and $x = y$ gives $y \\in y \\leftrightarrow y \\notin y$.\nThis is [[Russell's paradox|0HESMMSYEG]].\n[[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] replaces unrestricted comprehension by the separation scheme $\\exists y\\, \\forall x\\, (x \\in y \\leftrightarrow x \\in a \\wedge \\varphi(x))$, which only forms subsets of a given set $a$.\n",
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