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  "body": "\n## Problem\n\nWrite all of classical mathematics as a formal system with finitely describable axioms and rules of inference.\nThen prove, by finitary means only, that this formal system is consistent.\nHere a proof is finitary when it reasons only about finite strings of symbols in a concrete, checkable way.\nConsistency means that no formula $\\varphi$ is derivable together with its negation $\\neg \\varphi$.\n\n## Remarks\n\nThe program is the main form of [[formalism|46JCQ6KDA4]].\nIt also asked for a proof of completeness: every statement of the system should be provable or refutable.\nGödel's [[incompleteness theorems|X9WYNYE29C]] of 1931 showed that the program cannot be carried out as first intended.\nA consistent, effectively axiomatized system that contains enough arithmetic is incomplete.\nSuch a system also cannot prove its own consistency.\nIn 1936 Gerhard Gentzen proved the consistency of the [[Peano axioms|0QFT7Z76N5]] of first-order arithmetic, see [[Gentzen's consistency proof|QPV0HNP22S]].\nHis proof uses transfinite induction up to the ordinal $\\varepsilon_0$, which goes beyond the finitary means of the original program.\nPartial versions of the program are studied in [[proof theory|N4EDJNMTEF]] and in [[reverse mathematics|7SQS69T17R]].\n\n## Explanations\n\n### L1 human\n\nHilbert wanted to put all of mathematics on a safe base.\nFirst, write every theorem as the result of fixed rules applied to fixed axioms, like moves in a game.\nSecond, show with simple, finite reasoning that the game can never produce a contradiction such as $0 = 1$.\nGödel showed that the second step is impossible in the strong form that Hilbert hoped for.\n",
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