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    Made withinDC&Austin
    In the Mostowski model, the set A of rationals can be lin... — Carmelics
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    Home/Modality & Possibility
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    In the Mostowski model, the set A of rationals can be linearly ordered but cannot be well-ordered.

    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.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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 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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    Philosophy of Language1 linked

    Related

    Any model-theoretic demonstration of AC's failure is relative to a metatheory th...Diaconescu's theorem shows that the Axiom of Choice follows from the Law of Excl...If one adopts a constructivist framework following Brouwer or Bishop, the linear...In the Mostowski model, A = ℚ and H is the group of order-automorphisms of (ℚ, <...
    +5 moreShow less
    The Axiom of Choice holds for collections of non-empty finite sets in this model...The Mostowski model is constructed within ZF set theory, which itself presuppose...The claim therefore smuggles in classical assumptions about linear order that ar...The existing linear order on ℚ is preserved, but the Axiom of Choice fails in fo...Therefore, the claim that ℚ 'cannot' be well-ordered conflates model-internal im...

    Similar

    The existing linear order on ℚ is preserved, but the Axiom of Choice f...74%Shoham's theory characterizes S(K) as the set of models in K that are ...74%By Cohen's results, there exist models N and N' both satisfying the ax...74%Second-order logic therefore has sentences with models but no countabl...73%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    For more on the proof theory of second-order logic, see Buss (1998). There is a translation of many sorted logic further to single sorted first order logic due essentially to Herbrand (1930), see also Wang (1952) and Schmidt (1951). This can be used to obtain many of the basic properties of first order logic first for many sorted logic and then further for second-order logic with general models. The most important application of general models is the Completeness Theorem: Theorem 14 (Henkin 1
    Extraction notes

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

    Details

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