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 …

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 …

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 …

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

Oct 1, 2022 · just had some questions about countable and uncountable infinities. If we take a limit that results in ∞0 ∞ 0, we typically conclude that the limit is just ∞ ∞, correct?

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 …

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

5 $\omega_1$ is the first uncountable ordinal, or, equivalently, the set of all countable ordinals. The countable ordinals in turn can be constructed by the following rules: 0 is a countable …

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 …

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 …