My friend and I were discussing infinity and stuff about it and ran into some disagreements regarding countable and uncountable infinity. As far as I understand, the list of all natural numbers is
Lastly, we can always remove countable sets from uncountable sets and still end up with uncountable sets. For example, $ (\mathbb {R} \setminus \mathbb {N})$ is still an uncountable set.
set theory - What makes an uncountable set "uncountable"? - Mathematics ...
I'm trying to show that the interval $(0,1)$ is uncountable and I want to verify that my proof is correct My solution: Suppose by way of contradiction that $(0, 1)$ is countable. Then we can crea...
real analysis - Proving that the interval $ (0,1)$ is uncountable ...
The question is not well-posed because the notion of an infinite sum $\sum_ {\alpha\in A}x_\alpha$ over an uncountable collection has not been defined. The "infinite sums" familiar from analysis arise in the context of analyzing series defined by sequences indexed over $\mathbb {N}$, and the series is defined to be the limit of the partial sums. The only objects defined here are (1) finite ...
I once called a dense set an uncountable set. I was told this was wrong, as the set was dense, and not uncountable. I didn't have the mathematical knowledge to find this confusing, and instead thou...
Yes, the real numbers are uncountable, but your answer is circular. It does not explain why, it says that they are because the Cantor diagonal argument works, and it works because they are uncountable.
A simple way to see that the cantor set is uncountable is to observe that all numbers between $0$ and $1$ with ternary expansion consisting of only $0$ and $2$ are part of cantor set. Since there are uncountably many such sequences, so cantor set is uncountable.