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  "body": "\n## Summary\n\nProof theory is the branch of mathematical logic that treats [[proofs|0B94TKFQJX]] as mathematical objects.\nIt studies formal proofs in systems of [[predicate logic|086QKPF0A6]] and of arithmetic by combinatorial means.\nIt began with Hilbert's aim to prove the consistency of mathematics by finitary methods.\nIts main tools include normal forms of proofs, cut elimination and ordinal analysis.\n\n## Explanations\n\n### L1 human\n\nIn most of mathematics, a proof is a tool to reach a theorem.\nIn proof theory, the proof itself is the object of study, like a number or a graph.\nA formal proof is a finite tree of formulas built by fixed rules.\nSo one can count its steps, transform it, and ask what it can never reach.\nFor example, one can show that a system never proves $0 = 1$.\nOne way is to put every proof in a simple normal form, and to check that no proof in normal form ends with $0 = 1$.\n",
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