Math 2.0

Mathematical logic and foundations Ordinals and cardinals Theorem

Transfinite induction

Builds on Well-founded relationPredicate

Statement

Let RR be a well-founded relation on a set AA and let P(x)P(x) be a predicate on AA. Assume that, for each x∈Ax \in A, if P(y)P(y) holds for each y∈Ay \in A with yRxy \mathrel{R} x, then P(x)P(x) holds. Then P(x)P(x) holds for each x∈Ax \in A.

Remarks

Proof: if the set of xx with ¬P(x)\neg P(x) is nonempty, it has an RR-minimal element mm. Then P(y)P(y) holds for each yRmy \mathrel{R} m, so P(m)P(m) holds, a contradiction. For << on ℕ\mathbb{N}, this is strong induction. For ordinal numbers, one often splits the step into three cases: zero, successor ordinals and limit ordinals.

Reading path

To understand this page from the ground up, read these 18 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptSet, definition, no evidence levelSetdefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionAntisymmetric relation, definition, no evidence levelAntisymmetric relationdefinitionReflexive relation, definition, no evidence levelReflexive relationdefinitionTransitive relation, definition, no evidence levelTransitive relationdefinitionPreorder, definition, no evidence levelPreorderdefinitionPartial order, definition, no evidence levelPartial orderdefinitionMinimal element, definition, no evidence levelMinimal elementdefinitionWell-founded relation, definition, no evidence levelWell-founded relationdefinitionTransfinite induction, statement, no evidence levelTransfinite inductionstatement
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. Proposition concept
  3. Set definition
  4. Implication definition
  5. Predicate definition
  6. Ordered pair definition
  7. Universal quantifier definition
  8. Cartesian product definition
  9. Subset definition
  10. Binary relation definition
  11. Antisymmetric relation definition
  12. Reflexive relation definition
  13. Transitive relation definition
  14. Preorder definition
  15. Partial order definition
  16. Minimal element definition
  17. Well-founded relation definition
  18. Transfinite induction statement

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Transfinite_induction, lead section
About this page
Tag0ZHQMQ2C1N
URIm2://0ZHQMQ2C1N
Aliaseswell-founded-induction
Kindstatement, theorem
WritingW1 stated
MSC03E10
StatusThis copy of the site has no build data. Run m2 status 0ZHQMQ2C1N.
UsesWell-founded relation, Predicate
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