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    The order type Omega of the natural order on ordinal numb... — Carmelics
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    Home/Modality & Possibility
    HistoryEditSee Inverse

    The order type Omega of the natural order on ordinal numbers is itself one of the ordinal numbers

    Modality & PossibilityTruth & Knowledge
    ?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.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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 2 of 2
    ?
    • 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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    Modality & PossibilityTruth & Knowledge

    Related

    A genuine ordinal number must occupy the same ontological category as the ordina...Admitting Omega as an ordinal within On generates Burali-Forti paradox: Omega wo...Every set that is a well-ordering has an order typeFrege and Russell's logicist programs showed that unrestricted comprehension ove...
    +5 moreShow less
    In standard ZFC set theory, the class of all ordinals (On) is a proper class, no...NFU's stratification restrictions that make this move safe are not truth-preserv...NFU's type-level distinctions between sets and their order types mean Omega is a...Order types of well-orderings are ordinal numbersThe natural order on ordinal numbers is a set in NFU

    Similar

    There is a natural order on ordinal numbers93%The natural order on ordinal numbers is a set in NFU91%The natural order on ordinal numbers is a well-ordering and a set in N...90%There is no ordinal for the order type of the set of all ordinals.87%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
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    Ordinal numbers are defined as equivalence classes of well-orderings under similarity. There is a natural order on ordinal numbers, and in NFU as in the usual set theory it turns out to be a well-ordering—and, as in naive set theory, a set! Since the natural order on the ordinal numbers is a set, it has an order type \(\Omega\) which is itself one of the ordinal numbers. Now in the usual set theory we prove that the order type of the restriction of the natural order on the ordinals to the ordina
    Extraction notes

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

    Details

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