Cardinality is a way of measuring the size of a set. Two sets have the same cardinality if they are equinumerous, i.e. if there is a bijective mapping between them. A has a larger cardinality than B if there is an injective mapping from B to A. Cardinality gives a precise way to talk about different sizes of infinity.
This concept has the prerequisites:
- Define the relations of equinumerosity and dominance, and show that these are equivalence and order relations, respectively
- Prove the Schroeder-Bernstein Theorem: that if two sets each dominate each other, then they are equinumerous.
- Show that no set is equinumerous with its power set.
- Be able to manipulate cardinal numbers algebraically
- Be aware of the Continuum Hypothesis
Core resources (read/watch one of the following)
→ Elements of Set Theory
An introductory textbook on axiomatic set theory.
- Chapter 6, "Cardinal numbers and the Axiom of Choice," up through subsection "Ordering cardinal numbers," pages 128-150
- Chapter 6, subsections "Arithmetic of infinite cardinals" and "Continuum Hypothesis," pages 162-166
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