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    The restricted principle that all well-founded sets are w... — Carmelics
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    Home/Modality & Possibility
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    The restricted principle that all well-founded sets are well-orderable is consistent with this theory

    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.Models of the theory can be constructed in environments where Choice holds
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    • 2.In such constructions, well-founded sets inherit well-orderability
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Consistency relative to a model where Choice holds establishes only relative consistency, not absolute consistency of the restricted principle.
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    • 2.If the ambient theory used to construct such models itself requires large cardinal assumptions or AC, the argument is question-begging against anti-choice foundationalists like Solovay.
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    • 3.Solovay's 1970 model demonstrates that all sets of reals can be Lebesgue measurable assuming an inaccessible cardinal, undermining the claim that well-foundedness alone drives well-orderability.
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    Reason against 2 of 2
    ?
    • 1.The inference from 'well-founded sets inherit well-orderability in Choice-models' to 'the restricted principle is consistent' conflates semantic truth-in-a-model with syntactic consistency.
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    • 2.Diaconescu's theorem shows Choice can be derived from seemingly innocuous structural assumptions, meaning the supporting argument may smuggle in Choice rather than isolate well-foundedness as the operative factor.
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    Related

    Consistency relative to a model where Choice holds establishes only relative con...Diaconescu's theorem shows Choice can be derived from seemingly innocuous struct...If the ambient theory used to construct such models itself requires large cardin...In such constructions, well-founded sets inherit well-orderability
    +3 moreShow less
    Models of the theory can be constructed in environments where Choice holdsSolovay's 1970 model demonstrates that all sets of reals can be Lebesgue measura...The inference from 'well-founded sets inherit well-orderability in Choice-models...

    Similar

    Cantor's well-ordering principle states that every set can be put into...86%In such constructions, well-founded sets inherit well-orderability84%The well-ordering principle is equivalent to the Axiom of Choice78%The Cantor-Lawvere principle (CL) is a theorem of first-order logic, n...76%

    Source

    AI-extracted1/3 agreementValid
    SEP: settheory-alternative
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    The Axiom of Choice in any global form is inconsistent with this theory, but it is consistent for all well-founded sets to be well-orderable (in fact, this will be true in the models described above if the construction is carried out in an environment in which Choice is true). This is sufficient for the usual mathematical applications.
    Extraction notes

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

    Details

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