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    In standard ZFC set theory, the class of all ordinals (On... — Carmelics
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    Challenges→The order type Omega of the natural order on ordinal numbers is itself one of the ordinal numbers

    In standard ZFC set theory, the class of all ordinals (On) is a proper class, not a set, so it admits no order type as an ordinal.

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    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Cantor's Burali-Forti paradox shows that assuming On is a set leads to contradiction, so ZFC correctly excludes it.
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    • 2.Order types are defined as isomorphism classes of well-ordered sets; only sets can be members of such classes in ZFC.
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    • 3.Proper classes lack cardinality and cannot be ranked by ordinals without collapsing the cumulative hierarchy.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.One can coherently define an 'absolute order type' of On in extended frameworks (like Morse-Kelley set theory) without contradiction.
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    • 2.The restriction to sets for order types is a convention of ZFC, not a metaphysical fact about ordering or well-foundedness.
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    • 3.Proper classes possess well-ordering and transitivity; denying them an order type seems arbitrary rather than principled.
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    Truth & Knowledge1 linkedModality & Possibility1 linked

    Related

    Cantor's Burali-Forti paradox shows that assuming On is a set leads to contradic...One can coherently define an 'absolute order type' of On in extended frameworks ...Order types are defined as isomorphism classes of well-ordered sets; only sets c...Proper classes lack cardinality and cannot be ranked by ordinals without collaps...
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    Proper classes possess well-ordering and transitivity; denying them an order typ...The order type Omega of the natural order on ordinal numbers is itself one of th...The restriction to sets for order types is a convention of ZFC, not a metaphysic...

    Details

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