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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that The order type Omega of the natural order on ordinal numbers is itself one of the ordinal numbers

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.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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    • 2.Admitting Omega as an ordinal within On generates Burali-Forti paradox: Omega would be less than itself, since On is well-ordered by membership.
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    • 3.NFU's stratification restrictions that make this move safe are not truth-preserving translations of ZFC's intended ontology of ordinals.
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    Reason for 2 of 2
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    • 1.Frege and Russell's logicist programs showed that unrestricted comprehension over ordinal-generating principles yields contradictions without artificial type stratification.
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    • 2.NFU's type-level distinctions between sets and their order types mean Omega is an ordinal only in a deflated, type-shifted sense, not the same ordinal concept it purports to extend.
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    • 3.A genuine ordinal number must occupy the same ontological category as the ordinals it orders, which NFU's stratification systematically prevents.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The natural order on ordinal numbers is a set in NFU
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    • 2.Every set that is a well-ordering has an order type
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    • 3.Order types of well-orderings are ordinal numbers
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