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    The totality of all ordinal numbers cannot itself form a ... — Carmelics
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    Challenges→There exists a single universal list of all possible positions in well-ordered sets, called the ordinal numbers.

    The totality of all ordinal numbers cannot itself form a set, as shown by the Burali-Forti paradox of 1897.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.If all ordinals formed a set, that set would be an ordinal larger than itself, creating logical contradiction.
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    • 2.ZFC set theory avoids this paradox by distinguishing sets from proper classes, treating ordinals as the latter.
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    • 3.Unrestricted comprehension leads to contradiction; some mathematical collections cannot be sets without inconsistency.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The paradox only arises from informal reasoning; formal ZFC proves consistent without assuming all ordinals form a set.
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    • 2.Category theory and other frameworks handle collections of ordinals coherently without requiring the Burali-Forti argument.
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    • 3.The claim conflates a limitation of set theory axioms with a metaphysical truth about what collections 'can' be.
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    Connections

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    Proof of definition segments1 linkedModality & Possibility1 linked

    Related

    Category theory and other frameworks handle collections of ordinals coherently w...If all ordinals formed a set, that set would be an ordinal larger than itself, c...The claim conflates a limitation of set theory axioms with a metaphysical truth ...The paradox only arises from informal reasoning; formal ZFC proves consistent wi...
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    There exists a single universal list of all possible positions in well-ordered s...Unrestricted comprehension leads to contradiction; some mathematical collections...ZFC set theory avoids this paradox by distinguishing sets from proper classes, t...

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    claim
    Perspectives
    2 (1 for, 1 against)
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    1 edit