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 …

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 $\mathbb R$ …

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.

Sep 13, 2018 · In set theory, when we talk about the cardinality of a set we have notions of finite, countable and uncountably infinite sets. Main Question Let's talk about the dimension of a vector …

Oct 1, 2022 · 0 just had some questions about countable and uncountable infinities. If we take a limit that results in $\frac { \infty } {0}$, we typically conclude that the limit is just $\infty$, correct? But if …

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 …

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

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 …

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. …