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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 The semantics of a formal system rich enough to contain elementary mathematics cannot be fully defined in terms of mathematical functions within that same 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.Tarski's undefinability theorem shows truth is undefinable within a system, but this is consistent with truth being definable by a richer metalanguage that is itself mathematical.
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    • 2.The claim conflates the limits of object-language self-reference with the limits of mathematical semantics as such, since no system is required to define its own truth predicate.
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    • 3.Hartry Field's deflationary program demonstrates that semantic notions like truth can be reduced to purely formal, non-semantic primitives without invoking anything beyond mathematical structure.
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    Reason for 2 of 2
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    • 1.Gödel's incompleteness theorems concern provability within a fixed axiomatic system, not the expressive capacity of mathematical functions in general.
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    • 2.A formal system's semantics can be fully specified by a mathematical model in the set-theoretic sense, as Tarski's model theory demonstrates, without any non-mathematical remainder.
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    • 3.The supporting argument equivocates between 'definable within the system' and 'definable mathematically,' and rejecting the former does not entail rejecting the latter.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Any consistent formal system containing elementary arithmetic contains true statements that cannot be proved within the system.
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    • 2.A statement that is well-formed, meaningful, and truthful carries semantic information about the system.
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    • 3.If semantic information about a system exists that cannot be captured by provability within the system, then the system's semantics exceed what its internal mathematical functions can define.
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