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  "body": "\n## Summary\n\nThe foundations of mathematics study the basic framework in which all of mathematics is built.\nThey ask which language mathematics is written in, which axioms it starts from, and what counts as a [[proof|0B94TKFQJX]].\nToday the usual answer is [[predicate logic|086QKPF0A6]] together with the axioms of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].\nThe field also studies the limits of this framework, such as the [[incompleteness theorems|X9WYNYE29C]].\n\n## Explanations\n\n### L1 human\n\nEvery theorem rests on earlier theorems, and these rest on definitions and axioms.\nThe foundations of mathematics look at the bottom of this tower.\nFor example, the natural numbers can be built from sets: $0$ is the empty set, $1$ is $\\set{0}$, $2$ is $\\set{0, 1}$, and so on.\nThen facts about numbers become facts about sets, and only the axioms of set theory are needed.\nAround 1900, paradoxes in naive set theory made this question urgent, and [[logicism|5PGSQYB5Y1]], [[formalism|46JCQ6KDA4]] and [[intuitionism|XQHF63BBYF]] gave different answers.\n",
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