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

We can use the above theorem to show that $\mathbb R$ is in fact with bijection with $\mathcal P (\mathbb N)$, and therefore $\mathbb R$ is not countable. Since the above shows that $\mathbb R$ …

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...

May 16, 2024 · We know the interval $ [0, 1]$ is uncountable. You can think of the binary expansions of all numbers in $ [0, 1]$. This will give you an uncountable collection of sequences.

Dec 3, 2025 · There is something I don't understand about the proof that non-empty perfect sets are uncountable. The same proof is present in Rudin's Principles of Mathematical Analysis. Do we …

Jun 4, 2023 · And since $\aleph_0$ is the cardinality of any countable set, this means that this power set must be uncountable. Some other ways to construct infinite sets are simply to add elements to an …

6 Show that if a $\sigma$-algebra is infinite, that it contains a countably infinite collection of disjoint subsets. An immediate consequence is that the $\sigma$-algebra is uncountable.

Mar 28, 2019 · By construction, this sequence is different from all the others. This contradicts the assumption that this set of sequences is countable. Hence, $ [0,1]$ must be uncountable, and …

Jan 1, 2017 · Not to mention, it is far from useful to prove more complicated cardinalities and ones of actual mathematical interest. If you want to actually understand "cardinality" and countable vs. …

A set $A$ is countable if $A\approx\mathbb {N}$, and uncountable if it is neither finite nor countably infinite.