The ordinal numbers are a way of measuring the size of well-ordered sets. They give a way of talking about different sizes of infinity. They are needed to construct the cardinal numbers, which measure the sizes of sets more generally.
This concept has the prerequisites:
- natural numbers as sets (We must treat natural numbers as sets to use them as ordinal numbers.)
- well orderings (The ordinal numbers are well ordered, and defined using transfinite recursion.)
- set operations (The ordinal numbers are constructed with set union.)
- Define what it means for a set to be an ordinal
- Know that the ordinals can be divided into limit and successor ordinals
- Show that the subset relation is a well ordering on the ordinals
- Define what it means for two well-ordered sets to be isomorphic
- Show that every well-ordered set is isomorphic to a unique ordinal
- Show that for every ordinal, there is a larger ordinal
Core resources (read/watch one of the following)
→ Notes on Set Theory (2013)
Lecture notes for a course on axiomatic set theory.
→ A Course in Mathematical Logic
A graduate textbook in mathematical logic.
Location: Section 10.2, "Ordinals," pages 468-477
→ Elements of Set Theory
An introductory textbook on axiomatic set theory.
Location: Chapter 7, "Orderings and ordinals," subsections "Epsilon-images" through "Ordinal numbers," pages 182-194
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