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    Arithmetical truth cannot be defined in arithmetic — Carmelics
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    Arithmetical truth cannot be defined in arithmetic

    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
    ?
    • Attempting to define arithmetical truth in arithmetic leads to paradoxes such as the Liar paradox
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Tarski's undefinability theorem applies to classical first-order arithmetic, but non-standard or typed arithmetic systems may escape its scope.
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    • 2.Feferman's explicit mathematics and Kripke's theory of truth demonstrate that truth predicates can be consistently embedded in formal systems with careful stratification.
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    • 3.The impossibility result is therefore system-relative, not an absolute metaphysical constraint on the definability of arithmetical truth.
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    Reason against 2 of 2
    ?
    • 1.The Liar paradox arises from unrestricted self-reference, not from defining truth in arithmetic per se.
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    • 2.Restricted truth predicates that apply only to sentences of bounded complexity, as in Hájek and Pudlák's work on bounded arithmetic, avoid Liar-style paradoxes.
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    • 3.Thus the supporting argument conflates a pathology of unrestricted self-reference with a principled limitation on all arithmetical truth definitions.
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    Topics

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    Connections

    1 topic

    Philosophy of Language1 linked

    Related

    Attempting to define arithmetical truth in arithmetic leads to paradoxes such as...Feferman's explicit mathematics and Kripke's theory of truth demonstrate that tr...Restricted truth predicates that apply only to sentences of bounded complexity, ...Tarski's undefinability theorem applies to classical first-order arithmetic, but...
    +3 moreShow less
    The Liar paradox arises from unrestricted self-reference, not from defining trut...The impossibility result is therefore system-relative, not an absolute metaphysi...Thus the supporting argument conflates a pathology of unrestricted self-referenc...

    Similar

    Attempting to define arithmetical truth in arithmetic leads to paradox...82%The set of true arithmetical sentences cannot be defined in the langua...81%Every pure mathematical truth is a necessary truth.78%Sentences do not have truth conditions78%

    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