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    Given a well-ordering of any set (provable from AC itself... — Carmelics
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    Challenges→A symmetric function defined on a set of pairs cannot be a choice function on that set

    Given a well-ordering of any set (provable from AC itself), a canonical choice function exists that is not symmetric yet satisfies all formal requirements of a choice function.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.Well-orderings are constructive mathematical objects that break symmetry by definition, enabling canonical selections without appeal to arbitrary choice.
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    • 2.A choice function need only satisfy the formal requirement of selecting one element per set; symmetry is an aesthetic preference, not a logical requirement.
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    • 3.AC proves well-orderings exist; leveraging their asymmetry for canonical choice functions demonstrates AC's constructive potential beyond non-constructive existence claims.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Calling a choice function 'canonical' while admitting it lacks symmetry seems to conflate formal adequacy with philosophical canonicality—a category error.
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    • 2.AC's utility lies in proving *existence* of choice functions, not constructing specific ones; the claim overstates what AC provides.
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    • 3.Any well-ordering is dependent on arbitrary choices in its construction, making dependent choice functions equally non-canonical as the original problem.
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    Related

    A choice function need only satisfy the formal requirement of selecting one elem...A symmetric function defined on a set of pairs cannot be a choice function on th...AC proves well-orderings exist; leveraging their asymmetry for canonical choice ...AC's utility lies in proving *existence* of choice functions, not constructing s...
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    Any well-ordering is dependent on arbitrary choices in its construction, making ...Calling a choice function 'canonical' while admitting it lacks symmetry seems to...Well-orderings are constructive mathematical objects that break symmetry by defi...

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