Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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..

share|improve this question
This is valid, provided you replace $\mathbb R^1\in\mathbb R^*$ by $\mathbb R^1\subset\mathbb R^*$. –  Did Sep 10 '12 at 6:43
I see, that is clearer. –  James Sep 10 '12 at 6:44
Not clearer, true instead of false. –  Did Sep 10 '12 at 6:45
Is it because $\mathbb{R}^1$ is not an element? –  James Sep 10 '12 at 6:47
$\mathbb{R}^1$ is not an element of $\mathbb{R}^*$ as it is not finite. But it is a subset. –  Henry Sep 10 '12 at 6:49
show 1 more comment

1 Answer

up vote 0 down vote accepted

$\textbf{Hint}$ 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.

share|improve this answer
That proves the bullet "$\mathbb{R}$ is uncountable $\rightarrow \mathbb{R}^1$ is uncountable" but I think James already knew that. –  Henry Sep 10 '12 at 6:46
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.