Math 2.0

Mathematical logic and foundations Concept

System F

Builds on Simply typed lambda calculus

Summary

System F, also called the polymorphic lambda calculus or second-order lambda calculus, extends the simply typed lambda calculus with quantification over types. It was introduced independently by Jean-Yves Girard in logic and by John C. Reynolds in programming languages. A term of type ∀α. τ\forall \alpha.\, \tau works uniformly for every type α\alpha, which models generic programs. Girard proved that every typed term of System F is strongly normalizing, and under the Curry–Howard correspondence it matches second-order intuitionistic propositional logic.

Reading path

To understand this page from the ground up, read these 24 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLambda calculus, concept, no evidence levelLambda calculusconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementType theory, concept, no evidence levelType theoryconceptCurry–Howard correspondence, concept, no evidence levelCurry–Howard correspo…conceptSimply typed lambda calculus, concept, no evidence levelSimply typed lambda c…conceptSystem F, concept, no evidence levelSystem Fconcept
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. Element definition
  2. Intuitionistic logic concept
  3. Lambda calculus concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Mathematical proof concept
  15. Propositional logic concept
  16. Tautology definition
  17. Logical equivalence definition
  18. Double negation law statement
  19. Proof by contradiction concept
  20. Russell's paradox statement
  21. Type theory concept
  22. Curry–Howard correspondence concept
  23. Simply typed lambda calculus concept
  24. System F concept

Where this leads

6 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/System_F
About this page
TagSVNYKTGQ48
URIm2://SVNYKTGQ48
Kindconcept
WritingW0 stub
MSC03B40, 03B38, 68N18
UsesSimply typed lambda calculus
Same as
Changed3 November 2026

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