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  "body": "\n## Definition\n\nThe *foundations of mathematics* is the part of mathematical logic and philosophy that studies the basic framework of mathematics.\nIt studies:\n\n1. the formal languages in which mathematical statements are written, such as [[predicate logic|086QKPF0A6]];\n2. the axiom systems from which mathematical theorems are derived, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]];\n3. the notion of [[proof|0B94TKFQJX]], and the properties of the whole framework, such as its consistency and its completeness.\n\n## Remarks\n\nFor example, the [[Peano axioms|0QFT7Z76N5]] can be interpreted in set theory.\nEach natural number becomes a set, and each axiom becomes a theorem of set theory.\nA theorem about prime numbers alone, such as the infinitude of primes, is not a foundational result: it is about objects inside the framework, not about the framework.\nThe main schools of the early twentieth century were [[logicism|5PGSQYB5Y1]], [[formalism|46JCQ6KDA4]] and [[intuitionism|XQHF63BBYF]].\n",
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