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    Both the positive real numbers and the positive integers ... — Carmelics
    Home/Modality & Possibility
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    Supports→Not all infinite sets have the same cardinality.

    Both the positive real numbers and the positive integers are infinite sets.

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    Related propositions within the same area of thought.
    Not all infinite sets have the same cardinality.The positive real numbers have a greater cardinality than the positive integers.

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    The set of real numbers is infinite89%The positive real numbers have a greater cardinality than the positive...81%The set of prime numbers is infinite.79%Not all infinite sets have the same cardinality.76%

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    At this point, one might be forgiven for thinking that there is no cardinal number greater than that of the natural numbers, just as there is no extended real number larger than \(+\infty\). However, Cantor’s second striking result is that the cardinality of the positive real numbers is in fact greater than the cardinality of the positive integers, and his third striking result is that for every set, the set of all its subsets—its power set—has an even greater cardinality. Although many differen

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