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    A consistent formal system of arithmetic must contain sta... — Carmelics
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    Home/Modality & Possibility
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    A consistent formal system of arithmetic must contain statements that are true but not provable within that system.

    Modality & PossibilityTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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 against 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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    Modality & PossibilityTruth & Knowledge

    Connections

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    Philosophy of Language1 linked

    Related

    Arithmetic is consistent (contains no contradictions).Because the system is consistent, such a self-referential statement cannot be bo...By a construction related to the liar's paradox, one can construct a statement w...From a formalist or anti-realist perspective (Wittgenstein, Dummett), there is n...
    +4 moreShow less
    If we cannot prove consistency from within, the conditional 'if consistent, then...The claim that the Gödel sentence is 'true' presupposes a Platonist standard mod...The supporting argument assumes the consistency of arithmetic as a fixed, establ...Without a prior commitment to a privileged interpretation of arithmetic, 'true b...

    Similar

    Any consistent formal system containing elementary arithmetic contains...94%Any consistent formal system containing elementary arithmetic is funda...87%Any consistent formal system that contains arithmetic as a subsystem i...87%Gödel's 1931 proof demonstrated that any consistent formal system cont...87%

    Source

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    Since arithmetic is consistent this does not lead to paradoxes, but to incompleteness. By a construction related to the liars paradox Gödel proved that such a system must contain statements that are true but not provable: there are true sentences of the form “I am not provable”.
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit