GF4 Cardinality--Countable & Uncountable Sets

Bijections and Counting; Hotel Infinity, Countable and Uncountable Sets; Cantor's diagonal argument; Continuum Hypothesis. Subscribe @Shahriari for more undergraduate math videos.
00:00 Introduction
00:40 To count is to give a bijection
03:35 Bijections & "Size" of (possibly infinite) sets
05:04 Hotel Infinity
09:04 Definition: two sets with the "same cardinality"
10:34 Cardinality behaves like "size"
13:21 Definition: Finite, infinite, & countably infinite Sets
15:01 Definition: Countable & Uncountable Sets
15:40 Are the integers countable?
17:37 Are there any uncountable sets? Are the rationals uncountable?
18:24 Proof: The rationals are countable
22:06 Are there any uncountable sets? Are the reals uncountable?
22:28 Theorem: The interval [0,1] is uncountable
23:29 Proof: Cantor's diagonal argument
29:18 Discussion: Can you name many non-algebraic numbers in the interval [0, 1]?
31:36 The Continuum Hypthesis
33:21 The theorems of Gödel and Cohen

One of a series of lectures by Shahriar Shahriari on basic mathematical concepts used in undergraduate college mathematics.

The Series on Functions is as follows:
The vocabulary and the results are for general functions from one set to another (as opposed to just real valued functions on the real line) as encountered in courses in Linear Algebra, Combinatorics, Analysis, or Abstract Algebra.

Shahriar Shahriari is the William Polk Russell Professor of Mathematics at Pomona College in Claremont, CA, U.S.A.