A total ordering R on a set S is a well ordering if every subset of S has a smallest element. Well orderings are important because one can use a generalization of mathematical induction known as transfinite induction. The canonical example is the ordinal numbers.
This concept has the prerequisites:
- order relations (Well orderings are a subclass of order relations.)
- set operations
- natural numbers as sets (One must view natural numbers as sets to treat ordinary mathematical induction as a special case of transfinite induction.)
- Know what it means for a relation to be a well ordering
- Know the transfinite induction principle and why it is justified
- Be able to define functions using transfinite recursion
Core resources (read/watch one of the following)
→ Notes on Set Theory (2013)
Lecture notes for a course on axiomatic set theory.
→ Elements of Set Theory
An introductory textbook on axiomatic set theory.
Location: Chapter 7, "Orderings and ordinals," subsection "Well orderings," pages 172-178
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