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    Carmelics

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

    It is not the case that P ≠ NP is unlikely to be independent of strong formal theories such as PA or ZFC

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

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Gödelian incompleteness demonstrates that syntactic complexity of a statement's form is a poor guide to its provability within a given formal system.
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    • 2.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.
      ?

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    • 3.The absence of known independence proofs for P≠NP reflects our limited proof-theoretic tools, not the intrinsic provability of the statement within PA or ZFC.
      ?

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    Reason for 2 of 2
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    • 1.Scott Aaronson and others have shown that known proof techniques (algebrization, relativization, natural proofs) are formally blocked from resolving P≠NP, suggesting the statement resists standard mathematical machinery.
      ?

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    • 2.A statement that systematically evades all current proof strategies within a formal system is precisely the kind of statement independence results historically attach to, as Gödel's original incompleteness construction illustrates.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.P ≠ NP can be formulated as an arithmetical statement of the form ∀x∃y ψ(x,y) with only bounded numerical quantifiers
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    • 2.Statements of this logical form are generally believed not to be independent of theories like PA that approximate the mathematical axioms used in practice
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    • 3.We currently possess no reason to suspect that P ≠ NP is more likely to be independent of PA or ZFC than other currently open number-theoretic statements
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