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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
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    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that P ≠ NP is unlikely to be independent of Peano Arithmetic or ZFC.

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Gödel's incompleteness results show that arithmetical statements of low logical complexity can nonetheless be independent of PA, as demonstrated by Paris-Harrington.
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      Think about whether this reason is strong or weak

    • 2.The universality of P≠NP across computational models and its resistance to standard proof techniques mirrors structural features of known independence results.
      ?

      Think about whether this reason is strong or weak

    • 3.Epistemic familiarity with a statement's syntactic form provides no reliable evidence about its provability in a given formal system.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.Harvey Friedman's program demonstrates that combinatorial statements with no apparent set-theoretic content can require large cardinal axioms beyond ZFC for their proof.
      ?

      Think about whether this reason is strong or weak

    • 2.The informal consensus that P≠NP is 'probably' unprovable by current methods is epistemically compatible with, and may even suggest, genuine independence rather than mere technical difficulty.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.P ≠ NP is expressible as an arithmetical statement of the form ∀x∃y ψ(x,y) with only bounded numerical quantifiers.
      ?

      Think about whether this reason is strong or weak

    • 2.Such statements are generally believed unlikely to be independent of strong theories like PA.
      ?

      Think about whether this reason is strong or weak

    • 3.There is currently no reason to suspect P ≠ NP is more likely to be independent of PA than other open number-theoretic statements.
      ?

      Think about whether this reason is strong or weak

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    Strongest counterpoint
    Explore the most compelling reason on the other side.