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  • 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
  • real analysis - Proving that the interval $(0,1)$ is uncountable . . .
    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
  • 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
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  • 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
  • elementary set theory - What do finite, infinite, countable, not . . .
    The term countable is somewhat ambiguous (1) I would say that countable and countably infinite are the same That is, a set $A$ is countable (countably infinite) if
  • 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





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