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
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Alternative foundational systems such as von Neumann–Bernays–Gödel (NBG) set theory formally accommodate proper classes, including the class of all cardinals, as legitimate mathematical objects with determinate extensions.

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

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Proper classes lack determinate extensions since we cannot quantify over them or collect them into well-defined wholes, challenging claims of their objective existence.
      ?

      Think about whether this reason is strong or weak

    • 2.The formal accommodation of proper classes solves no mathematical problems ZFC cannot solve through alternative methods, making it ontologically extravagant without epistemic gain.
      ?

      Think about whether this reason is strong or weak

    • 3.NBG's consistency relative to ZFC suggests proper classes are merely convenient notational devices, not genuinely existing mathematical objects with intrinsic reality.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.NBG formally distinguishes sets from proper classes, allowing consistent treatment of collections like all cardinals that cannot be sets without contradiction.
      ?

      Think about whether this reason is strong or weak

    • 2.Proper classes in NBG have well-defined membership conditions and satisfy the axioms governing their behavior, making them mathematically rigorous objects.
      ?

      Think about whether this reason is strong or weak

    • 3.NBG's framework enables elegant expression of mathematical truths (e.g., about cardinal arithmetic) that ZFC can only awkwardly formalize or must reformulate.
      ?

      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.
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42