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    Harvey Friedman's work on independence results shows that... — Carmelics
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    Challenges→P ≠ NP is unlikely to be independent of strong formal theories such as PA or ZFC

    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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    Reasons For

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    • 1.Friedman's TREE(3) and related functions provably exceed PA's proof-theoretic strength, demonstrating genuine independence of simple combinatorial statements.
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    • 2.Independence results show that PA's axioms don't capture all truths about finite structures, revealing fundamental limits of first-order arithmetic.
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    • 3.Arithmetically simple statements can express deep mathematical content; syntactic simplicity doesn't entail semantic or proof-theoretic simplicity.
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    Reasons Against

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    Reason against
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    • 1.Most of Friedman's independence results require non-standard graph/tree encodings; truly 'simple' arithmetic statements remain PA-provable.
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    • 2.The philosophical significance of independence via large cardinals or ordinal analysis is contested; it may reflect proof method limitations, not mathematical reality.
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    • 3.Determining whether a statement is 'arithmetically simple' involves subjective judgments about form; the claim conflates technical independence with conceptual simplicity.
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    Related

    Arithmetically simple statements can express deep mathematical content; syntacti...Determining whether a statement is 'arithmetically simple' involves subjective j...Friedman's TREE(3) and related functions provably exceed PA's proof-theoretic st...Independence results show that PA's axioms don't capture all truths about finite...
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    Most of Friedman's independence results require non-standard graph/tree encoding...P ≠ NP is unlikely to be independent of strong formal theories such as PA or ZFCThe philosophical significance of independence via large cardinals or ordinal an...

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