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    Made withinDC&Austin
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    Under AFA, ℘* and ℘* are distinct — Carmelics
    Home/Modality & Possibility
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    Under AFA, ℘* and ℘* are distinct

    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.AFA (Anti-Foundation Axiom) permits non-wellfounded sets
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    • 2.Under AFA, ℘* equals WF (well-founded sets only) while ℘* equals V (the full universe including non-wellfounded sets)
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Under ZFC with the Foundation Axiom, V=WF, so ℘* and ℘* collapse into the same object, making their distinctness a feature of AFA alone.
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    • 2.A claim about distinctness that holds only relative to a chosen axiomatic system cannot be treated as an absolute ontological fact about sets.
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    • 3.Benacerraf's structuralist insight entails that multiple incompatible set-theoretic universes can equally well serve as foundations, undermining privileging AFA's verdict.
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    Reason against 2 of 2
    ?
    • 1.Boolos and others in the iterative conception tradition argue that the cumulative hierarchy exhausts the genuinely coherent notion of set, making WF=V non-negotiable.
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    • 2.If WF=V is conceptually mandatory, then ℘* and ℘* are necessarily co-extensional, and their alleged distinctness under AFA reflects notational inflation rather than real mathematical difference.
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    Related

    A claim about distinctness that holds only relative to a chosen axiomatic system...AFA (Anti-Foundation Axiom) permits non-wellfounded setsBenacerraf's structuralist insight entails that multiple incompatible set-theore...Boolos and others in the iterative conception tradition argue that the cumulativ...
    +3 moreShow less
    If WF=V is conceptually mandatory, then ℘* and ℘* are necessarily co-extensional...Under AFA, ℘* equals WF (well-founded sets only) while ℘* equals V (the full uni...Under ZFC with the Foundation Axiom, V=WF, so ℘* and ℘* collapse into the same o...

    Similar

    Oscar and Oscar− are distinct at t'91%Oscar and Oscar− are distinct at t91%Therefore Rab and Rba would be identical, which contradicts their dist...87%Non-Identity: L3 is numerically distinct from both L1 and L2.87%

    Source

    AI-extracted1/3 agreementValid
    SEP: nonwellfounded-set-theory
    View source passageHide passage
    Assuming FA, the fixed points are unique; frequently they are the empty set. With AFA, the greatest fixed points usually have non-wellfounded members. We shall study this in more detail when we turn to coalgebra. For now, we return to the last of the example equations at the top of this section, V   =  ℘V. This equation has no solutions in sets due to Cantor’s Theorem. However, in terms of classes, this equation does have solutions, as we know. The universal class V is a solution
    Extraction notes

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

    Details

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