Class 19: Uncountable Sets

### Schedule

**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 $S$, $|pow(S)| = 2^{|S|}$.

# #

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

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

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

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## 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?

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

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

# # #

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

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

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

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

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?