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    Carmelics

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    Home/Original/inverse
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    Inverse View

    It is not the case that The restricted principle that all well-founded sets are well-orderable is consistent with this theory

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 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 for 2 of 2
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    • 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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    Reasons Against

    1 perspective
    Reason against
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    • 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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