3.5. The continuum

The size of the set of real numbers, $\mathbb{R}$ is denoted by $\mathfrak{c}:=|\mathbb{R}|$. Cantor’s diagonal argument implies that ${\aleph }_0<\mathfrak{c}$. In this section I prove the stronger result that $2^{{\aleph }_0}=\mathfrak{c}$.

Here is a list of lemmas students proved in class that culminate in the main result of the section. Recall that the closed interval is $[a,b]:=\{x\in \mathbb{R}:a\le x\le b\}$, and the half-open interval $[a,b):=\{x\in \mathbb{R}:a\le x<b\}$, and similarly for other types of intervals.

Lemma.

For all reals $a<b$, $|[0,1]|=|[a,b]|$.

Lemma.

$|[0,1]|=|[0,1)|$.

Theorem.

$$|\mathbb{R}|=|\mathcal{P}(\mathbb{N})|$$

Proof:

Consider $f:[0,1)\to \mathcal{P}(\mathbb{N})$ given by $f(0.a_0{a}_1\dots a_k\dots ):=\{a_{k}10^k+1:k\in \mathbb{N}\}$, where the preimage is given by the decimal expansion of the real number without repeating 9’s. The rest is left to the reader.

Exercises

Exercises.

Prove that the following sets all have size $\mathfrak{c}$.

1. The set of infinite binary sequences.

Hint.

The solution is found in these notes.

2. The set of infinite sequences of real numbers.
3. The Cantor set. (Will be formally defined in Topology.)
4. The subsets of a countably infinite set.
5. The infinite subsets of a countably infinite set.
6. The countable subsets of $\mathbb{R}$.

Hard exercises.

1. If $A$ is countable and $|B|\le \mathfrak{c}$, there are at most $\mathfrak{c}$-many functions $A\to B$.
2. The set of continuous functions $\mathbb{R}\to \mathbb{R}$ has size $\mathfrak{c}$, and the set of discontinuous functions $\mathbb{R}\to \mathbb{R}$ has size $2^{\mathfrak{c}}$.
3. The product of countably many sets of size $\le \mathfrak{c}$ has size $\le \mathfrak{c}$.