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 $2^{{\aleph }_0}=\mathfrak{c}$.

The Cantor set

Note:

This set is best understood when constructed via sketches and examples. This is done in lectures.

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.
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.

7. If $A$ is countable and $|B|\le \mathfrak{c}$, there are at most $\mathfrak{c}$-many functions $A\to B$.
8. 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 $2^{\mathfrak{c}}$.
9. The product of countably many sets of size $\le \mathfrak{c}$ has size $\le \mathfrak{c}$.