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    There exists a single universal list of all possible posi... — Carmelics
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    Home/Modality & Possibility
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    There exists a single universal list of all possible positions in well-ordered sets, called the ordinal numbers.

    Modality & PossibilityProof of definition segments
    ?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.All positions of one well-ordering must correspond to initial positions in any other well-ordering.
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    • 2.The correspondence of positions across all well-ordered sets is total and consistent.
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    • 3.A single sequence beginning with first, second, third, and so on, can enumerate all such positions.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The totality of all ordinal numbers cannot itself form a set, as shown by the Burali-Forti paradox of 1897.
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    • 2.Any purported 'universal list' of ordinals would itself occupy an ordinal position, generating a strictly larger ordinal beyond the list.
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    • 3.A collection that cannot be a completed set cannot serve as a well-defined universal enumeration of positions.
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    Reason against 2 of 2
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    • 1.In predicativist foundations (Weyl, Feferman), only those ordinals definable from prior stages without impredicative totalities are mathematically legitimate.
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    • 2.A 'single universal list' of all ordinals presupposes the completed infinite totality of transfinite positions, which predicativism rejects as circular self-constitution.
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    • 3.Without an independent justification for impredicative set formation, the claim oversteps what constructively grounded mathematics can sanction.
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    Modality & PossibilityProof of definition segments

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    Related

    A 'single universal list' of all ordinals presupposes the completed infinite tot...A collection that cannot be a completed set cannot serve as a well-defined unive...A single sequence beginning with first, second, third, and so on, can enumerate ...All positions of one well-ordering must correspond to initial positions in any o...
    +5 moreShow less
    Any purported 'universal list' of ordinals would itself occupy an ordinal positi...In predicativist foundations (Weyl, Feferman), only those ordinals definable fro...The correspondence of positions across all well-ordered sets is total and consis...The totality of all ordinal numbers cannot itself form a set, as shown by the Bu...Without an independent justification for impredicative set formation, the claim ...

    Similar

    If there were an ordinal corresponding to the order type of the set of...82%There is no ordinal for the order type of the set of all ordinals.81%Order types of well-orderings are ordinal numbers81%The natural order on ordinal numbers is a well-ordering and a set in N...80%

    Source

    AI-extracted1/3 agreementValid
    SEP: infinity
    View source passageHide passage
    Cantor noted that for any two well-ordered sets, the initial positions in one ordering (the first, the second, the third, etc.) correspond to the initial positions in the other, the way that they do for finite sets. In fact, he showed that all of the positions of one well-ordering must correspond to initial positions in the other. (If this weren’t true, then the set of positions in one that don’t correspond to positions in the other would be non-empty for each set, and the first elements of thes
    Extraction notes

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

    Details

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