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. …

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 $\mathbb R$ …

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

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 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 …

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 …

Nov 5, 2023 · Whether the uncountable sum of zeros is zero or not simply depends on the definition of uncountable sum you're using. After all, concepts in mathematics require formal definitions to be …

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.

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 …

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 ordinal If …