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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that 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.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
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    • 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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      Think about whether this reason is strong or weak

    • 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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      Think about whether this reason is strong or weak

    • 3.Proper classes possess well-ordering and transitivity; denying them an order type seems arbitrary rather than principled.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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      Think about whether this reason is strong or weak

    • 2.Order types are defined as isomorphism classes of well-ordered sets; only sets can be members of such classes in ZFC.
      ?

      Think about whether this reason is strong or weak

    • 3.Proper classes lack cardinality and cannot be ranked by ordinals without collapsing the cumulative hierarchy.
      ?

      Think about whether this reason is strong or weak

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