Cardinality

Equivalent Sets

Finite Sets

Finite Cardinalities

N

Denumerable Sets

N ~ 2N

So, what gives?

Denumerable + 1 = Denumerable

Listable Sets?

Union of Denumerable Sets

Z is denumerable

Q is denumerable

Q is denumerable

Countable Sets

Countable Union

Uncountable Sets

Uncountable Sets

Example of the Construction

c

c

Order of Cardinal Numbers

Cardinal Numbers

Cantor's Theorem

Consequences

Schröder-Bernstein Theorem

c

The Axiom of Choice

The Comparability Theorem

Banach-Tarski Paradox

Theorem 5.36

The Continuum Hypothesis

The Continuum Hypothesis