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    If the power set of the set of all sets were a subset of ... — Carmelics
    Home/Modality & Possibility
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    Supports→The set-theoretical universe cannot form a set.

    If the power set of the set of all sets were a subset of the set of all sets, then the power set of the set of all sets would not have a larger cardinality than the set of all sets.

    Modality & PossibilityTruth & Knowledge
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    Modality & PossibilityTruth & Knowledge

    Key Terms

    Set of all sets(as used in set theory and logic)
    A hypothetical collection that would contain every possible set in existence. This concept leads to logical problems, which is why mathematicians generally avoid treating it as a real object.
    Subset(as used in logic and set theory)
    A smaller group that is completely contained within a larger group—for example, all the red books are a subset of all the books in the library.
    cardinality

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    Related propositions within the same area of thought.
    (Central to comparing infinite sets and establishing that no universal set exists.)
    A measure of the size of a set, indicating the number of elements it contains.
    power set(Used in Cantor's theorem: every set has cardinality strictly less than its power set.)
    The set of all subsets of a given set.
    set(von Neumann's class theory)
    A class that is a member (of some class).

    Related

    Cantor's Theorem states that the power set of any given set has a larger cardina...If the set-theoretical universe formed a set (the set of all sets), then the pow...The set-theoretical universe cannot form a set.This contradicts Cantor's Theorem, which guarantees the power set has strictly g...

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    If the set-theoretical universe formed a set (the set of all sets), th...89%Cantor's Theorem states that the power set of any given set has a larg...88%Therefore such a set would be at least as large as its own power set, ...84%This contradicts Cantor's Theorem, which guarantees the power set has ...84%

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    SEP: philosophy-mathematics
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    One question that has been important from the beginning of set theory concerns the difference between sets and proper classes. (This question has a natural counterpart for category theory: the difference between small and large categories.) Cantor’s diagonal argument forces us to recognize that the set-theoretical universe as a whole cannot be regarded as a set. Cantor’s Theorem shows that the power set (i.e., the set of all subsets) of any given set has a larger cardinality than the given set i

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