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    Diaconescu's theorem shows that the Axiom of Choice follo... — Carmelics
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    Challenges→In the Mostowski model, the set A of rationals can be linearly ordered but cannot be well-ordered.

    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.

    ?Rate how convincing each reason is below to see the overall strength.
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    Reasons For

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    Reason for
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    • 1.Diaconescu's theorem rigorously demonstrates that LEM implies AC in constructive type theory, revealing framework-specific logical dependencies.
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    • 2.Classical logic and intuitionistic logic treat these principles differently, showing their relationship is indeed dependent on foundational assumptions.
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    • 3.Topos theory provides a unified framework where axiom interactions become explicit, validating the claim that entanglement is framework-dependent.
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    Reasons Against

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    Reason against
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    • 1.The implication LEM→AC holds in constructive settings, but this reflects proof-theoretic artifacts rather than deep philosophical entanglement.
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    • 2.Calling their relationship 'framework-dependent' obscures that AC and LEM remain conceptually distinct principles across all standard foundations.
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    • 3.The theorem's insight concerns formal derivability, not whether these axioms are genuinely entangled in mathematical practice or ontology.
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    Related

    Calling their relationship 'framework-dependent' obscures that AC and LEM remain...Classical logic and intuitionistic logic treat these principles differently, sho...Diaconescu's theorem rigorously demonstrates that LEM implies AC in constructive...In the Mostowski model, the set A of rationals can be linearly ordered but canno...
    +3 moreShow less
    The implication LEM→AC holds in constructive settings, but this reflects proof-t...The theorem's insight concerns formal derivability, not whether these axioms are...Topos theory provides a unified framework where axiom interactions become explic...

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    2 (1 for, 1 against)
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