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    The map T on isomorphism types of well-founded extensiona... — Carmelics
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    Perspectives
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    Home/Truth & Knowledge
    HistoryEditSee Inverse

    The map T on isomorphism types of well-founded extensional relations is not a set function.

    Proof of definition segmentsTruth & 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 map T sends the isomorphism type of a relation R to the isomorphism type of R^iota, which is one type higher in terms of TST.
      ?

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    • 2.There is no reason to believe that T is a function in the usual set-theoretic sense.
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    • 3.Considerations from the Burali-Forti paradox provide grounds for this conclusion.
      ?

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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.In NFU and related systems, the Burali-Forti paradox is blocked by stratification, allowing T to be defined as a legitimate set function.
      ?

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    • 2.Jensen's consistency proof of NFU demonstrates that type-lowering permits ordinal arithmetic without paradox, undermining the claim's generality.
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    • 3.The claim implicitly assumes a ZF-style cumulative hierarchy, but alternative foundations dissolve rather than inherit the Burali-Forti obstruction.
      ?

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    Reason against 2 of 2
    ?
    • 1.Scott's trick in ZFC assigns sets to isomorphism types by selecting rank-minimal representatives, making T a well-defined set function within standard set theory.
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    • 2.If Scott's trick is applicable, the non-functionality of T is an artifact of naive type-ascent rather than a deep feature of well-founded extensional relations.
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    Topics

    Truth & KnowledgeProof of definition segments

    Connections

    2 topics

    Skepticism2 linkedPhilosophy of Language1 linked

    Related

    Considerations from the Burali-Forti paradox provide grounds for this conclusion...If Scott's trick is applicable, the non-functionality of T is an artifact of nai...In NFU and related systems, the Burali-Forti paradox is blocked by stratificatio...Jensen's consistency proof of NFU demonstrates that type-lowering permits ordina...
    +4 moreShow less
    Scott's trick in ZFC assigns sets to isomorphism types by selecting rank-minimal...The claim implicitly assumes a ZF-style cumulative hierarchy, but alternative fo...The map T sends the isomorphism type of a relation R to the isomorphism type of ...

    Similar

    This endomorphism is induced by the map T on isomorphism types of well...83%A structure isomorphic to models of NFU can be constructed in the isom...83%The map T sends the isomorphism type of a relation R to the isomorphis...80%Ordinals in NFU are isomorphism types of relations79%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
    View source passageHide passage
    In any model of NFU, a structure which looks just like one of these models can be constructed in the isomorphism classes of well-founded extensional relations. The theory of isomorphism classes of well-founded extensional relations with a top element looks like the theory of (an initial segment of) the usual cumulative hierarchy, because every set in Zermelo-style set theory is uniquely determined by the isomorphism type of the restriction of the membership relation to its transitive closure. Th
    Extraction notes

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

    Details

    There is no reason to believe that T is a function in the usual set-theoretic se...
    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit