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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
  • 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
  • Why Are the Reals Uncountable? - Mathematics Stack Exchange
    Why are the reals uncountable? I've been talking with a friend about why the answer of these three questions can be the case when they somewhat seem to contradict each other I seek clarification on the subject Herein lies a summary of the discussion: Person A: By way of Cantor Diagonalization it can be shown that the reals are uncountable
  • 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
  • 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
  • Does an uncountable discrete subspace of the reals exist?
    For example there is no uncountable discrete subspace of the irrationals The reals are a second-countable space, so any subspace is also second-countable, which prevents the subspace from having an uncountable discrete subspace
  • Proving a set is uncountable - Mathematics Stack Exchange
    A set $A$ is countable if $A\approx\mathbb {N}$, and uncountable if it is neither finite nor countably infinite
  • 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





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