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 There exists a nonstandard model M in which all polynomial-time computable functions are total but the exponential function is not total.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.The completeness theorem guarantees a model exists but not that it is ω-consistent; nonstandard models may satisfy sentences false in the standard interpretation.
      ?

      Think about whether this reason is strong or weak

    • 2.Kreisel's squeezing argument shows that provability in formal systems and truth in intended models can come apart, so model existence from consistency does not vindicate the claim about actual computational totality.
      ?

      Think about whether this reason is strong or weak

    • 3.The polynomial-time functions being 'total' in M means only that M satisfies their defining axioms, not that they are total over the genuine natural numbers, making the claim modal-epistemically ambiguous.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Putnam's model-theoretic argument establishes that formal consistency alone cannot fix intended reference, so 'polynomial-time computable' in M may not refer to the same class as in the standard model.
      ?

      Think about whether this reason is strong or weak

    • 2.If the exponential function fails to be total in M, then M's arithmetic is too weak to interpret standard complexity-theoretic distinctions, undermining the claim that M's polynomial-time class is the complexity-theoretically relevant one.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.The theory S^1_2 + ∃y¬∃z ε(2,y,z) is proof-theoretically consistent.
      ?

      Think about whether this reason is strong or weak

    • 2.By the completeness theorem for first-order logic, every consistent theory has a model.
      ?

      Think about whether this reason is strong or weak

    • 3.In such a model M, there exists an element a satisfying ¬∃z ε(2,a,z), meaning 2^a does not exist in M.
      ?

      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.