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 Harvey Friedman's work on independence results shows that arithmetically simple statements (e.g., finite Ramsey variants) can be independent of PA despite their Π₂ or Σ₁ form.

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

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Most of Friedman's independence results require non-standard graph/tree encodings; truly 'simple' arithmetic statements remain PA-provable.
      ?

      Think about whether this reason is strong or weak

    • 2.The philosophical significance of independence via large cardinals or ordinal analysis is contested; it may reflect proof method limitations, not mathematical reality.
      ?

      Think about whether this reason is strong or weak

    • 3.Determining whether a statement is 'arithmetically simple' involves subjective judgments about form; the claim conflates technical independence with conceptual simplicity.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Friedman's TREE(3) and related functions provably exceed PA's proof-theoretic strength, demonstrating genuine independence of simple combinatorial statements.
      ?

      Think about whether this reason is strong or weak

    • 2.Independence results show that PA's axioms don't capture all truths about finite structures, revealing fundamental limits of first-order arithmetic.
      ?

      Think about whether this reason is strong or weak

    • 3.Arithmetically simple statements can express deep mathematical content; syntactic simplicity doesn't entail semantic or proof-theoretic simplicity.
      ?

      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.