Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that A symmetric function defined on a set of pairs cannot be a choice function on that set

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.The argument presupposes that symmetry over all permutations is required, but choice functions need only be definable relative to a fixed well-ordering of the domain.
      ?

      Think about whether this reason is strong or weak

    • 2.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.
      ?

      Think about whether this reason is strong or weak

    • 3.Thus the argument establishes only that no *symmetric* procedure suffices, which is precisely what motivates AC as an independent axiom, not a refutation of choice functions per se.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Fraenkel's permutation models show symmetry-based impossibility results hold in set theories with urelements but do not straightforwardly transfer to pure ZF set theory.
      ?

      Think about whether this reason is strong or weak

    • 2.In ZF without urelements, sets have no 'intrinsic' symmetric indistinguishability of the kind the argument exploits, since all sets are distinguished by their membership structure.
      ?

      Think about whether this reason is strong or weak

    • 3.Therefore the permutation argument, following Fraenkel-Mostowski methodology, demonstrates the independence of AC from weaker systems rather than the impossibility of choice functions in standard set theory.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.For a symmetric function f defined on P, there exists a finite list L of pairs from P such that fixing all elements of pairs in L suffices to fix f and all its values
      ?

      Think about whether this reason is strong or weak

    • 2.For any pair U in P but not in L, a permutation π can be found that fixes all elements of pairs in L but does not fix the members of U
      ?

      Think about whether this reason is strong or weak

    • 3.Because π fixes all elements of pairs in L, π must fix the value of f at U
      ?

      Think about whether this reason is strong or weak

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.