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  "body": "\n## Definition\n\nA *formal system* consists of a formal language, a set of axioms, and a finite set of inference rules.\nA *formal proof* in such a system is a finite sequence or tree of formulas.\nIn it, each formula is an axiom or follows from earlier formulas by an inference rule.\n*Proof theory* is the branch of mathematical logic whose objects of study are formal proofs.\nThe proofs are taken in systems of [[predicate logic|086QKPF0A6]] and in theories such as arithmetic.\nIt studies their structure, their transformations and their length, and what follows for the theories in which they are written.\n\n## Remarks\n\nExample: the question whether every proof in the sequent calculus can be turned into a proof without the cut rule is a question of proof theory.\nIts answer is the cut-elimination theorem of Gentzen.\nNon-example: the question whether a given group is abelian is not a question of proof theory, because its object is a group and not a formal proof.\nMain parts of the field are structural proof theory, ordinal analysis, provability logic, reverse mathematics and proof complexity.\nThe informal notion that formal proofs model is [[mathematical proof|0B94TKFQJX]].\n",
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