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    "summary": "Henkin proved completeness of a simple type calculus for general models whose function domains satisfy the interpretation closure conditions, and constructed countable models of consistent theories in the stated language.\n\nHigher-order logic can quantify over functions and sets. If these variables must range over every mathematically possible function or set, a recursive proof system cannot capture all valid statements in the expressive calculus considered here. Henkin changed the semantic setting: general models may have smaller function domains, but those domains must still interpret every formula and satisfy the required closure conditions. His 1950 paper constructs such models from consistent sets of formulas and proves that validity in all general models matches formal provability. It also derives compactness and discusses nonstandard models of arithmetic. The distinction matters: a general model is not an arbitrary collection of functions, and the proof does not supply an effective procedure for deciding every formula. The countable-language result does not say all mathematical structures are countable. Gödel retains credit for first-order completeness and incompleteness; Church for the type calculus; and Henkin’s earlier 1949 proof remains a separate predecessor. This entry concerns the 1950 higher-type model construction and its stated consequences.",
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        "description": "Completeness is for the stated type theory with general/Henkin models, whose higher-type domains need not contain every set-theoretic function. It is not completeness for unrestricted full second-order semantics. The syntax, comprehension and model assumptions matter, and consistency is not a computational decision procedure for every theory.",
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      "statement": "# Henkin: complete proof rules for general models of type theory\n\nHenkin proved completeness of a simple type calculus for general models whose function domains satisfy the interpretation closure conditions, and constructed countable models of consistent theories in the stated language.\n\nHigher-order logic can quantify over functions and sets. If these variables must range over every mathematically possible function or set, a recursive proof system cannot capture all valid statements in the expressive calculus considered here. Henkin changed the semantic setting: general models may have smaller function domains, but those domains must still interpret every formula and satisfy the required closure conditions. His 1950 paper constructs such models from consistent sets of formulas and proves that validity in all general models matches formal provability. It also derives compactness and discusses nonstandard models of arithmetic. The distinction matters: a general model is not an arbitrary collection of functions, and the proof does not supply an effective procedure for deciding every formula. The countable-language result does not say all mathematical structures are countable. Gödel retains credit for first-order completeness and incompleteness; Church for the type calculus; and Henkin’s earlier 1949 proof remains a separate predecessor. This entry concerns the 1950 higher-type model construction and its stated consequences.\n\n## Scientific scope\n\nCompleteness is for the stated type theory with general/Henkin models, whose higher-type domains need not contain every set-theoretic function. It is not completeness for unrestricted full second-order semantics. The syntax, comprehension and model assumptions matter, and consistency is not a computational decision procedure for every theory.\n\n## People and systems\n\nLeon Henkin\n\n\n\n## Original sources\n\n- [Henkin,CompletenessintheTheoryofTypes,JSL15(2),81-91(1950),JSTOR2266967 originalfacsimile](https://aaronxyliu.github.io/download/PL/Henkin.pdf)\n",
      "summary": "Henkin proved completeness of a simple type calculus for general models whose function domains satisfy the interpretation closure conditions, and constructed countable models of consistent theories in the stated language.\n\nHigher-order logic can quantify over functions and sets. If these variables must range over every mathematically possible function or set, a recursive proof system cannot capture all valid statements in the expressive calculus considered here. Henkin changed the semantic setting: general models may have smaller function domains, but those domains must still interpret every formula and satisfy the required closure conditions. His 1950 paper constructs such models from consistent sets of formulas and proves that validity in all general models matches formal provability. It also derives compactness and discusses nonstandard models of arithmetic. The distinction matters: a general model is not an arbitrary collection of functions, and the proof does not supply an effective procedure for deciding every formula. The countable-language result does not say all mathematical structures are countable. Gödel retains credit for first-order completeness and incompleteness; Church for the type calculus; and Henkin’s earlier 1949 proof remains a separate predecessor. This entry concerns the 1950 higher-type model construction and its stated consequences.",
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