Uncountable vs Countable Infinity - Mathematics Stack Exchange 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 countably infinite and the list of reals between 0 and 1 is uncountably infinite
set theory - What makes an uncountable set uncountable? - Mathematics . . . The question in its current form is a bit unclear, but I'll try to address your concern below Assuming we are working with the axioms of $\mathsf {ZFC}$, we can construct an uncountable set by taking the power set of any set with infinitely many elements (e g $\mathbb{N}, \mathbb{Z}, \mathbb{R}$, etc )
Uncountable $\sigma$-algebra - Mathematics Stack Exchange As a secondary question: I think this result is supposed to be used to show that if $\mathcal A$ is a $\sigma$-algebra with infinitely many elements, then it's uncountable, but I wasn't able to show that the property mentioned above (the one I'm trying to show) was satisfied in this case
set theory - Union of Uncountably Many Uncountable Sets - Mathematics . . . By definition, uncountable means the set is not countable There are no other choices So, if you take 1 or more uncountable sets, it will stay in the biggest class, uncountable Even if you take uncountably many sets that are uncountable, there's no where above uncountable to go Uncountable isn't a cardinal
Proving a set is uncountable - Mathematics Stack Exchange Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers
The sum of an uncountable number of positive numbers 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
Why do we not need measures to be uncountably additive? As long as uncountable many sets have positive measure the union will have infinite measure (unlike for a countable sum) Thanks for any help and suggestions! Edit: I know that in the end uncountable additivity would imply that every subset of $\mathbb{R}$ has Lebesgue measure $0$, but I feel like the author of the notes has something else in