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 Sentences proved from first-order axioms are true in all models of those axioms, including countable models and non-standard models

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.The Löwenheim-Skolem theorem entails that first-order axioms for real analysis have countable models, yet 'true in all models' cannot capture intended mathematical truth about uncountable reals.
      ?

      Think about whether this reason is strong or weak

    • 2.Skolem's paradox reveals that 'truth in all models' is model-relative: a sentence about uncountability can be true in the intended model yet satisfied in a countable model via different satisfaction relations.
      ?

      Think about whether this reason is strong or weak

    • 3.Therefore, provability from first-order axioms tracks formal satisfiability across structures, not truth about the mathematical domain the axioms were designed to characterize.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Non-standard models of Peano arithmetic, as demonstrated by Thoralf Skolem in 1933, contain 'natural numbers' with no standard correlates, making first-order provability insufficient to exclude pathological interpretations.
      ?

      Think about whether this reason is strong or weak

    • 2.If sentences proved from first-order arithmetic axioms are 'true' in non-standard models containing infinite natural numbers, then 'true in all models' conflates formal truth-in-a-structure with arithmetical truth about the natural numbers.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.First-order logic satisfies the Completeness Theorem
      ?

      Think about whether this reason is strong or weak

    • 2.The Completeness Theorem uses the concept of a first-order structure, so provability implies truth across all such structures
      ?

      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.