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 Limitation of Size implies the Axiom of Choice

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.The derivation assumes the Axiom of Replacement to establish that the von Neumann ordinals constitute a proper class via Burali-Forti.
      ?

      Think about whether this reason is strong or weak

    • 2.The Axiom of Limitation of Size, as formulated by von Neumann, is itself equivalent to Replacement plus Choice, making the implication circular.
      ?

      Think about whether this reason is strong or weak

    • 3.Cantor-von Neumann set theories that restrict Replacement can model Limitation of Size without generating a class-sized ordinal spine sufficient to well-order the universe.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Michael Hallett and Akihiro Kanamori argue that 'same size' for proper classes requires a prior notion of class cardinality that is not definable without Choice-like assumptions.
      ?

      Think about whether this reason is strong or weak

    • 2.If class bijections require definable functions, the step from equinumerosity with the ordinals to a global well-ordering presupposes definable choice functions, making the argument question-begging.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The von Neumann ordinals form a proper class, essentially by the Burali-Forti paradox
      ?

      Think about whether this reason is strong or weak

    • 2.Because the von Neumann ordinals form a proper class, the universe must be the same size as the class of ordinals
      ?

      Think about whether this reason is strong or weak

    • 3.A class bijection between the universe and the ordinals can be used to define a global well-ordering of the universe
      ?

      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.