Math 2.0

Mathematical logic and foundations Definition

Proof theory

Builds on Mathematical proofPredicate logic

Definition

A formal system consists of a formal language, a set of axioms, and a finite set of inference rules. A formal proof in such a system is a finite sequence or tree of formulas. In it, each formula is an axiom or follows from earlier formulas by an inference rule. Proof theory is the branch of mathematical logic whose objects of study are formal proofs. The proofs are taken in systems of predicate logic and in theories such as arithmetic. It studies their structure, their transformations and their length, and what follows for the theories in which they are written.

Remarks

Example: 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. Its answer is the cut-elimination theorem of Gentzen. Non-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. Main parts of the field are structural proof theory, ordinal analysis, provability logic, reverse mathematics and proof complexity. The informal notion that formal proofs model is mathematical proof.

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To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Proposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptPredicate logic, concept, no evidence levelPredicate logicconceptProof theory, definition, no evidence levelProof theorydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Proposition concept
  2. Conjunction definition
  3. Disjunction definition
  4. Implication definition
  5. Negation definition
  6. Predicate definition
  7. Biconditional definition
  8. Modus ponens concept
  9. Mathematical proof concept
  10. Propositional logic concept
  11. Predicate logic concept
  12. Proof theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Proof_theory
About this page
TagTH8EGSCFYY
URIm2://TH8EGSCFYY
Kinddefinition
WritingW0 stub
MSC03F03, 03F05
DefinesProof theory
UsesMathematical proof, Predicate logic
Changed2 November 2026

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