Math 2.0

Mathematical logic and foundations Concept

Inductive type

Builds on Type theoryMathematical inductionLeast fixed point

Summary

An inductive type is a type given by a list of constructors. Its elements are exactly the values built by finitely many uses of these constructors. The natural numbers, with the constructors zero and successor, are the basic example; lists and trees are others. Each inductive type comes with a principle of recursion and of induction that defines functions and proves statements by cases on the constructors. Inductive types are the main way to introduce data in type theories such as Martin-Löf type theory and the calculus of inductive constructions.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionLeast fixed point, concept, no evidence levelLeast fixed pointconceptMathematical induction, concept, no evidence levelMathematical inductionconceptProposition, 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 theoryconceptInductive type, concept, no evidence levelInductive typeconcept
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. Least fixed point concept
  3. Mathematical induction 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. Inductive type concept

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Inductive_type
About this page
TagQ80JWCWNKX
URIm2://Q80JWCWNKX
Kindconcept
WritingW0 stub
MSC03B38, 68N15
UsesType theory, Mathematical induction, Least fixed point
Same as
Changed3 November 2026

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