Epsilon-induction
Sign in to saveIn set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction.
~20 min read
Encyclopedic overview
20 sectionsContents
- Statement
- In terms of classes
- Related notions of induction
- Ordinals
- Well-founded relations
- For negative predicates
- Infinite descending chains
- Self-membership
- Contrapositive
- Classical equivalents
- Disjunctive form
- Relation to regularity
- History
- Set induction from regularity and transitive sets
- Transitive set existence
- Comparison of epsilon and natural number induction
- Classical equivalents
- Least number principle
- See also
- Reference
In set theory, \in-induction, also called epsilon-induction or set-induction, is a principle that can be used to prove that all sets satisfy a given property. Considered as an axiomatic principle, it is called the axiom schema of set induction.
The principle implies transfinite induction and recursion. It may also be studied in a general context of induction on well-founded relations.
Excerpted from Wikipedia’s “Epsilon-induction” article, available under the CC BY-SA 4.0 licence.