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    There must be true arithmetical sentences which are not p... — Carmelics
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    There must be true arithmetical sentences which are not provable

    Truth & 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.The set of provable sentences in arithmetic and the set of true arithmetical sentences cannot coincide
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    • 2.All provable sentences are true
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Truth in arithmetic is not a language-independent fact but is defined relative to a model, making 'true but unprovable' model-relative rather than absolute.
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    • 2.Gödel sentences are true only in the standard model; in non-standard models of arithmetic, those same sentences can be false, undermining claims of absolute unprovable truth.
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    • 3.Hilbert's formalist position holds that mathematical truth just is provability within a formal system, so 'true but unprovable' collapses into a category error.
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    Reason against 2 of 2
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    • 1.P2 assumes soundness of the formal system, but if we cannot verify soundness without a stronger system, the argument begs the question against a strict formalist.
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    • 2.Lucas and Putnam's debates show that attributing truth to Gödel sentences requires adopting a Platonist semantic framework that is itself philosophically contested, not a neutral starting point.
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    Related

    All provable sentences are trueGödel sentences are true only in the standard model; in non-standard models of a...Hilbert's formalist position holds that mathematical truth just is provability w...Lucas and Putnam's debates show that attributing truth to Gödel sentences requir...
    +3 moreShow less
    P2 assumes soundness of the formal system, but if we cannot verify soundness wit...The set of provable sentences in arithmetic and the set of true arithmetical sen...Truth in arithmetic is not a language-independent fact but is defined relative t...

    Similar

    The set of provable sentences in arithmetic and the set of true arithm...90%The set of true arithmetical sentences cannot be defined in the langua...89%The set of sentences provable in arithmetic can be defined in the lang...85%By the Completeness Theorem, provable sentences are valid, but not all...82%

    Source

    AI-extracted1/3 agreementValid
    SEP: goedel-incompleteness
    View source passageHide passage
    Be that as it may, it seems that Gödel actually arrived at the first exact observations about incompleteness via a different route, during his attempts to contribute to Hilbert’s program, and not to undermine it (see Dawson 1997: Ch. IV). Namely, in 1930, Gödel made an effort to advance Hilbert’s program by attempting to prove the consistency of analysis (or, second-order arithmetic) with the resources of arithmetic, and thus reduce the consistency of the former to the consistency of the latter.
    Extraction notes

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

    Details

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