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 The Axiom of Choice is innocent in the context of second-order logic and is generally accepted in the second-order logic literature.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Second-order logic's quantification over all subsets is a semantic stipulation, not an ontological guarantee that choice functions exist.
      ?

      Think about whether this reason is strong or weak

    • 2.The Axiom of Choice asserts the existence of a specific function, which requires more than mere acknowledgment that subsets can be quantified over.
      ?

      Think about whether this reason is strong or weak

    • 3.Boolos and others have shown that the 'full' second-order semantics presupposes a background set theory that itself requires AC, making the 'innocence' claim circular.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Constructivist and predicativist traditions (Weyl, Feferman) reject impredicative quantification over all subsets as ontologically illegitimate.
      ?

      Think about whether this reason is strong or weak

    • 2.If second-order quantification is restricted to predicatively definable properties, as Feferman's program demands, AC is not automatically licensed.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The basic tenet of second-order logic is that all properties of elements of a fixed domain exist and can be quantified over.
      ?

      Think about whether this reason is strong or weak

    • 2.In set-theoretical terms, all subsets—definable or not—of a set of any cardinality exist and can be quantified over.
      ?

      Think about whether this reason is strong or weak

    • 3.Quantifying over all subsets naturally licenses the Axiom of Choice for families of subsets of any cardinality.
      ?

      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.