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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Arithmetical truth cannot be defined in arithmetic

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    Reasons For

    2 perspectives
    Reason for 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 for 2 of 2
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    • 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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    Reasons Against

    1 perspective
    Reason against
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    • Attempting to define arithmetical truth in arithmetic leads to paradoxes such as the Liar paradox
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