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  "body": "\n## Definition\n\n*Set theory* is the branch of mathematical logic whose objects of study are [[sets|0SJ3RX9TXA]] and the membership relation $\\in$ between an [[element|0EZ758XE73]] and a set.\nAn *axiomatic set theory* is a theory in [[first-order logic|086QKPF0A6]] with axioms about $\\in$.\nIts language has the binary relation symbol $\\in$, and possibly further symbols.\nThe standard axiomatic set theory is ZFC, [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with the axiom of choice.\n\n## Remarks\n\nExample: ZFC is an axiomatic set theory.\nIts language has the single relation symbol $\\in$, and its axioms, such as the [[axiom of extensionality|0TB16T2C97]], speak only about $\\in$ and equality.\nNon-example: Peano arithmetic is not a set theory in this sense, because its language has $0$, successor, addition and multiplication, and its objects are natural numbers.\nOther axiomatic set theories include Zermelo set theory, von Neumann–Bernays–Gödel set theory, Morse–Kelley set theory and Quine's New Foundations.\nQuestions such as the [[continuum hypothesis|0MN2MER9WN]] belong to set theory and are independent of ZFC, if ZFC is consistent.\n",
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