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  • set theory - What makes an uncountable set uncountable? - Mathematics . . .
    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 existing set by taking the union of an arbitrary set and known uncountable sets
  • Proof that a non-empty perfect set is uncountable
    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 assume that our
  • 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
  • Uncountable Summation of Zeros - Mathematics Stack Exchange
    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 rigorous, and there is no rule other than courtesy saying that these definitions conform to any sort of common sense or colloquial meaning
  • Whats the difference between a dense set and an uncountable set?
    The problem with an uncountable set, like the set of real numbers, is that finite subsets (that include all members between the lower and upper limits) don't exist $| [x,z]| = \infty \ \forall x,z \in \Bbb R$ There's an infinite number of members between any two members, and thus all subsets are infinite, which makes counting impossible (hence
  • Why is the Cantor set uncountable - Mathematics Stack Exchange
    A simple way to see that the cantor set is uncountable is to observe that all numbers between $0$ and $1$ with ternary expansion consisting of only $0$ and $2$ are part of cantor set Since there are uncountably many such sequences, so cantor set is uncountable
  • Proving that R is uncountable - Mathematics Stack Exchange
    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 therefore $\mathbb {R}$ must be uncountable
  • 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
  • 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 is the set of all real numbers uncountable?
    The number which is the diagonal is transformed s t it doesn't share the first digit of the first number nor the second digit with the second and so on Thus the number is unique to the list This is why Cantor's diagonal as a method proves the result that the reals are uncountable





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