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    Carmelics

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    LoyalLoyalJusticeJustice
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    Home/Original/inverse
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    Inverse View

    It is not the case that A general choice function over all non-empty sets requires selecting from collections with no computable or definable structure.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
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    • 1.For practical or constructive mathematics, choice functions only need apply to sets we can actually describe or enumerate.
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    • 2.The claim conflates logical existence (via AC) with epistemological availability—a choice function needn't be constructible to exist abstractly.
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    • 3.Many standard choice functions (lexicographic, cardinality-based) *do* rely on definable structure, weakening the necessity claim.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The Axiom of Choice applies to arbitrary collections, including those lacking definable well-orderings or computable selection rules.
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    • 2.Many infinite sets (e.g., uncountable ordinals) have no recursive structure enabling mechanical selection from their powersets.
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    • 3.If choice functions required computability, they'd fail for most transfinite collections—contradicting AC's logical independence from ZF.
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