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    If the metatheory is weakened to predicative or intuition... — Carmelics
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    Challenges→Strong completeness holds for many-sorted logic: if Γ ⊨ φ then Γ ⊢ φ

    If the metatheory is weakened to predicative or intuitionistic set theory, the Henkin model construction fails because maximal consistent extensions require non-constructive choice principles.

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    Key Terms

    Henkin model construction(as used in logic)
    A mathematical method (named after logician Leon Henkin) for building models that satisfy logical formulas by systematically adding elements to make sentences true.
    Intuitionistic set theory(as used in mathematical logic)
    A version of set theory that doesn't assume something is either true or false—instead, you have to be able to construct or prove something to say it exists.
    Maximal consistent extensions(as used in logic)
    The largest possible set of logical statements you can add to a theory without creating any contradictions.
    Metatheory(as the foundational framework underlying a logical system)
    The background rules and assumptions you need to use in order to study or reason about a logical system—like the scaffolding needed to build a house.

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    Non-constructive choice principles(as used in mathematical logic and set theory)
    Mathematical tools that let you make selections from infinite collections without having to explicitly describe how you're making each choice; they're powerful but somewhat abstract and 'non-constructive' because you can't always show your work step-by-step.
    Predicative(as used in set theory)
    A restriction in logic where you can only define something using things that already exist, not by referring to a larger set that includes the thing you're defining.

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    Proof of definition segments1 linkedPhilosophy of Language1 linked

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    Strong completeness holds for many-sorted logic: if Γ ⊨ φ then Γ ⊢ φ

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