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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 A consistent formal system of arithmetic must contain statements that are true but not provable within that system.

    ?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.The claim that the Gödel sentence is 'true' presupposes a Platonist standard model of arithmetic (ℕ) against which truth is assessed.
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    • 2.From a formalist or anti-realist perspective (Wittgenstein, Dummett), there is no model-independent notion of arithmetic truth beyond provability.
      ?

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    • 3.Without a prior commitment to a privileged interpretation of arithmetic, 'true but unprovable' collapses into 'unprovable in this system but provable in an extension'—a far weaker claim.
      ?

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    Reason for 2 of 2
    ?
    • 1.The supporting argument assumes the consistency of arithmetic as a fixed, established fact, but Gödel's second incompleteness theorem entails this consistency cannot itself be proven within the system.
      ?

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    • 2.If we cannot prove consistency from within, the conditional 'if consistent, then the Gödel sentence is true' provides no categorical grounds for asserting the sentence's truth—only its truth relative to an assumed but unverifiable hypothesis.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Arithmetic is consistent (contains no contradictions).
      ?

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    • 2.By a construction related to the liar's paradox, one can construct a statement within arithmetic that asserts 'I am not provable'.
      ?

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    • 3.Because the system is consistent, such a self-referential statement cannot be both provable and false — if it were provable, it would be false, yielding a contradiction.
      ?

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    Strongest counterpoint
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