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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 apparent bijection x ↦ {x} between ℘₁(V) and V cannot be a set in NFU.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Stratification is a syntactic criterion, but the existence of a function as a set is a semantic question about the model's domain.
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    • 2.Holmes and others have shown NFU admits non-standard models where some unstratified operations have set-realizations under alternative interpretations.
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    • 3.The failure of x↦{x} to be a set is a feature of standard NFU models, not a logical necessity derivable from stratification alone.
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    Reason for 2 of 2
    ?
    • 1.Quine's original NF and Jensen's NFU differ in that NFU's urelements create asymmetry: V and ℘₁(V) can have different cardinalities, making the bijection's non-existence a cardinality fact, not merely a stratification fact.
      ?

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    • 2.Presenting stratification failure as the explanation conflates the proof-theoretic reason for unprovability with the model-theoretic reason for the bijection's non-existence as a set.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The map x ↦ {x} has an unstratified definition.
      ?

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    • 2.In NFU, only stratified definitions give rise to sets.
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    • 3.There is no expectation that an unstratified definition yields a set.
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    Strongest counterpoint
    Explore the most compelling reason on the other side.