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    The set-theoretical universe cannot form a set. — Carmelics
    Home/Modality & Possibility
    HistoryEditSee Inverse

    The set-theoretical universe cannot form a set.

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Cantor's Theorem states that the power set of any given set has a larger cardinality than that set itself.
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    • 2.If the set-theoretical universe formed a set (the set of all sets), then the power set of the set of all sets would have to be a subset of the set of all sets.
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    • 3.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.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Cantor's Theorem presupposes the iterative conception of sets, which is one axiomatization among several legitimate alternatives.
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    • 2.Paraconsistent set theories (e.g., Routley's and Brady's) formally accommodate a universal set without explosive contradiction.
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    • 3.If a consistent paraconsistent framework permits a set of all sets, the impossibility claim is relative to classical logic, not absolute.
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    Reason against 2 of 2
    ?
    • 1.Quine's NF (New Foundations) is a consistent set theory in which a universal set exists without violating any theorem of that system.
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    • 2.The argument from Cantor's Theorem assumes ZFC-style comprehension, but NF restricts comprehension via stratification, blocking the paradox differently.
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    • 3.A claim that the set-theoretical universe 'cannot' form a set is therefore system-relative, not a metaphysically necessary truth.
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    Topics

    Modality & Possibility

    Connections

    1 topic

    Truth & Knowledge4 linked

    Related

    A claim that the set-theoretical universe 'cannot' form a set is therefore syste...Cantor's Theorem presupposes the iterative conception of sets, which is one axio...Cantor's Theorem states that the power set of any given set has a larger cardina...If a consistent paraconsistent framework permits a set of all sets, the impossib...
    +6 moreShow less
    If the power set of the set of all sets were a subset of the set of all sets, th...If the set-theoretical universe formed a set (the set of all sets), then the pow...Paraconsistent set theories (e.g., Routley's and Brady's) formally accommodate a...Quine's NF (New Foundations) is a consistent set theory in which a universal set...The argument from Cantor's Theorem assumes ZFC-style comprehension, but NF restr...This contradicts Cantor's Theorem, which guarantees the power set has strictly g...

    Similar

    V (the universe of all sets) is not a set89%There is a single universe of sets.86%If the set-theoretical universe formed a set (the set of all sets), th...85%There are many distinct concepts of set, each determining its own univ...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: philosophy-mathematics
    View source passageHide passage
    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
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit