Discrete Mathematics

Class 19: Uncountable Sets


Problem Set 8 is due Friday (4 November) at 6:29pm.

Friday, 4 November at 11am (Rotunda Dome Room): Steve Huffman (BSCS 2005, co-founder and CEO of Reddit), Computer Science Distinguished Alumni Speaker.

Exam 2 will be in class on Thursday, 10 November. See Class 18 notes for details.

Countable and Uncountable Sets

Definition. A set $S$ is countably infinite if and only if there exists a bijection between $S$ and $\mathbb{N}$.

Definition. A set $S$ is uncountable, if there exists no bijection between $S$ and $\mathbb{N}$.

The power set of $A$ ($\textrm{pow}(A)$)is the set of all subsets of $A$: $$ B \in \textrm{pow}(A) \iff B \subseteq A. $$

For all finite sets $S$, $|pow(S)| = 2^{|S|}$.

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For all sets $S$, $|pow(S)| > |S|$.


Prove $pow(\mathbb{N})$ is uncountable.


$\text{bitstrings} = \forall n \in \mathbb{N} . {0, 1}^n$.


Ordinal and Cardinal Numbers

$\omega$ is the smallest infinite ordinal. The first ordinal after $0, 1, 2, \cdots$.

What is the difference between an ordinal and cardinal number?


What should $2\omega$ mean?


Is $\text{InfiniteBitStrings} = {0, 1}^\omega$ countable?


Prove the number of real numbers in the interval $[0, 1]$ is uncountable.

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What set is bigger than $\mathbb{R}$?


Aleph-naught: $\aleph_0 = |\mathbb{N}|$ is the smallest infinite cardinal number.

How is $\omega$ different from $\aleph_0$?


$$2^{\aleph_0} = |pow(\mathbb{N})| = | [0, 1] | = | \mathbb{R} | = | \text{InfiniteBitStrings} | $$

What is necessary to conclude that it is not possible to settle the question of whether $\aleph_1 = 2^{\aleph_0}$ with the ZFC axioms?