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    Conflating syntactic unprovability with semantic model-th... — Carmelics
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    Challenges→Any sufficiently strong formal theory F satisfying the conditions of the first incompleteness theorem must possess non-standard models in addition to its intended standard model.

    Conflating syntactic unprovability with semantic model-theoretic multiplicity commits a use-mention error: G_F being unprovable in F does not entail F lacks a unique intended interpretation, only that F cannot prove it.

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    Key Terms

    G_F(Constructed via the diagonal lemma; satisfies equivalence (G))
    The Gödel sentence for system F — a sentence that encodes its own unprovability in F
    Multiplicity(one of the two opposing concepts being discussed)
    The quality of being many things or having multiple parts; the opposite of unity.
    Syntactic(as used in linguistics)
    Related to the rules and structure of how words are arranged in language, rather than their meaning.
    Unprovability(describing mathematical claims that cannot be established as true)
    The quality of a statement that cannot be logically proven true using the accepted rules and facts of a system, like a math problem that has no solution.
    intended interpretation(SQML and KQML semantics)

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    An interpretation for an applied modal language comprising the actual and merely possible individuals and worlds the language is understood to be about
    model-theoretic(mathematical logic)
    Related to the study of how logical symbols and formulas correspond to real structures and meanings (rather than just abstract rules).
    semantic(describing the level of word meaning)
    Relating to the meaning of words and sentences.
    use-mention error(fundamental logic error in arguments)
    A logical mistake where you confuse using a word with talking about the word itself—like saying 'dog has three letters' when you meant the word 'dog' has three letters.

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    2 topics

    Truth & Knowledge1 linkedModality & Possibility1 linked

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    Any sufficiently strong formal theory F satisfying the conditions of the first i...

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