Show all finite subsets of reals is uncountably infinite (or is it?).
Firstly, I assumed that "all finite subsets of reals" is equivalent to the Kleene closure of $\mathbb{R}$, $$\mathbb{R}^* = \mathbb{R}^0\cup\mathbb{R}^1\cup\mathbb{R}^2\cup…$$
- $\mathbb{R}$ is uncountable. $\Rightarrow \mathbb{R}^1$ is uncountable.
- $\mathbb{R}^1 \subset \mathbb{R}^* \Rightarrow \mathbb{R}^*$ is uncountable because the union of an uncountable set with another set is also uncountable.
- $\mathbb{R}^*$ is uncountably infinite.
Is this a valid proof? I am sort of new to the subject of proofs..
Best Answer
$\textbf{Hint}$ Singletons are finite subsets. How many singleton subsets of $\mathbb{R}$ are there?
The Kleene closure is a way of computing all the finite subsets but introducing it is not exactly necessary since the main idea really is $\mathbb{R}^1 \subset \mathbb{R}^*$, which is exactly the hint.