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    A choice function exists in constructive mathematics — Carmelics
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    A choice function exists in constructive mathematics

    Philosophy of LanguageTruth & 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.A choice is implied by the very meaning of existence in constructive mathematics
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    • 2.To constructively assert existence is to possess a procedure that produces a witness
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Diaconescu's theorem shows that the Axiom of Choice implies the Law of Excluded Middle in intuitionistic set theory (IZF).
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    • 2.Constructive mathematics explicitly rejects the Law of Excluded Middle as a valid logical principle.
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    • 3.Therefore, unrestricted choice functions in constructive mathematics generate a contradiction with its foundational logical commitments.
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    Reason against 2 of 2
    ?
    • 1.Bishop-style constructivism permits only dependent choice and countable choice, not full AC over arbitrary sets.
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    • 2.A general choice function over all non-empty sets requires selecting from collections with no computable or definable structure.
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    • 3.Possessing a witness-producing procedure for existence claims does not guarantee a uniform procedure across infinitely many arbitrary sets simultaneously.
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    Related

    A choice is implied by the very meaning of existence in constructive mathematicsA general choice function over all non-empty sets requires selecting from collec...Bishop-style constructivism permits only dependent choice and countable choice, ...Constructive mathematics explicitly rejects the Law of Excluded Middle as a vali...
    +4 moreShow less
    Diaconescu's theorem shows that the Axiom of Choice implies the Law of Excluded ...Possessing a witness-producing procedure for existence claims does not guarantee...Therefore, unrestricted choice functions in constructive mathematics generate a ...To constructively assert existence is to possess a procedure that produces a wit...

    Similar

    A choice is implied by the very meaning of existence in constructive m...90%Since f is a choice function on P and U is in P, f(U) must equal c or ...76%Predicativism is a compromise between classical and constructive viewp...73%If a choice function f on P were in Sym(V), then f would have a finite...73%

    Source

    AI-extracted1/3 agreementValid
    SEP: axiom-choice
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    The fact that the Axiom of Choice implies Excluded Middle seems at first sight to be at variance with the fact that the former is often taken as a valid principle in systems of constructive mathematics governed by intuitionistic logic, e.g. Bishop’s Constructive Analysis[16] and Martin-Löf’s Constructive Type Theory[17], in which Excluded Middle is not affirmed. In Bishop’s words, “A choice function exists in constructive mathematics because a choice is implied by the very meaning of existe
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    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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