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    The operation T on ordinal numbers can be defined in NFU — Carmelics
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    The operation T on ordinal numbers can be defined in NFU

    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.Ordinals in NFU are isomorphism types of relations
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    • 2.The operation T is definable on isomorphism types of relations
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.NFU's stratification requirements impose type-level constraints that block naive application of T to its own ordinal representatives.
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    • 2.The operation T as classically defined requires quantification over all ordinals at a single type level, which violates NFU's stratification.
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    Reason against 2 of 2
    ?
    • 1.Isomorphism types in NFU are sets of relations of a given type-level, making cross-level ordinal operations like T type-theoretically ill-formed.
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    • 2.Holmes's own reconstruction of T in NFU requires auxiliary lemmas about type-raising that are not part of the original classical definition, making the claim equivocal.
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    Related

    Holmes's own reconstruction of T in NFU requires auxiliary lemmas about type-rai...Isomorphism types in NFU are sets of relations of a given type-level, making cro...NFU's stratification requirements impose type-level constraints that block naive...Ordinals in NFU are isomorphism types of relations
    +2 moreShow less
    The operation T as classically defined requires quantification over all ordinals...The operation T is definable on isomorphism types of relations

    Similar

    There is a natural order on ordinal numbers81%The definition of ordinal numbers ensures that for any non-empty set o...80%Every cardinal number can be represented by an ordinal number78%The order type Omega of the natural order on ordinal numbers is itself...78%

    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