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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that In the Mostowski model, the set A of rationals can be linearly ordered but cannot be well-ordered.

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    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The Mostowski model is constructed within ZF set theory, which itself presupposes classical logic and standard set-theoretic semantics.
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    • 2.Any model-theoretic demonstration of AC's failure is relative to a metatheory that may itself require choice-like principles for its own consistency proofs.
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    • 3.Therefore, the claim that ℚ 'cannot' be well-ordered conflates model-internal impossibility with absolute metaphysical impossibility across all set-theoretic universes.
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    Reason for 2 of 2
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    • 1.Diaconescu's theorem shows that the Axiom of Choice follows from the Law of Excluded Middle in topos-theoretic foundations, suggesting their entanglement is framework-dependent.
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    • 2.If one adopts a constructivist framework following Brouwer or Bishop, the linear orderability of ℚ itself requires revision, undermining the asymmetry the Mostowski model purports to demonstrate.
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    • 3.The claim therefore smuggles in classical assumptions about linear order that are not neutral across foundational frameworks, making its modal force framework-relative rather than absolute.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.In the Mostowski model, A = ℚ and H is the group of order-automorphisms of (ℚ, <).
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    • 2.The existing linear order on ℚ is preserved, but the Axiom of Choice fails in forms strong enough to yield a well-ordering.
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    • 3.The Axiom of Choice holds for collections of non-empty finite sets in this model.
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