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# General models, descriptions, and choice in type theory

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In [4] Alonzo Church introduced an elegant and expressive formulation of type theory with λ-conversion. In [8] Henkin introduced the concept of a general model for this system, such that a sentence A is a theorem if and only if it is true in all general models. The crucial clause in Henkin's definition of a general model ℳ is that for each assignment φ of values in ℳ to variables and for each wff A, there must be an appropriate value of A in ℳ. Hintikka points out in [10, p. 3] that this constitutes a rather strong requirement concerning the structure of a general model. Henkin draws attention to the problem of constructing nonstandard models for the theory of types in [9, p. 324].

We shall use a simple idea of combinatory logic to find a characterization of general models which does not directly refer to wffs, and which is easier to work with in certain contexts. This characterization can be applied, with appropriate minor and obvious modifications, to a variety of formulations of type theory with λ-conversion. We shall be concerned with a language ℒ with extensionality in which there is no description or selection operator, and in which (for convenience) the sole primitive logical constants are the equality symbols Qoαα for each type α.

References
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[1]Andrews, Peter B., A reduction of the axioms for the theory of propositional types, Fundamenta Mathematicae, vol. 52 (1963), pp. 345350.
[2]Andrews, Peter B., A transfinite type theory with type variables, North-Holland, Amsterdam, 1965, 143 pp.
[3]Andrews, Peter B., Resolution in type theory, this Journal, vol. 36 (1971), pp. 414432.
[4]Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 5668.
[5]Church, Alonzo, Non-normal truth-tables for the propositional calculus, Boletin de la Sociedad Matematica Mexicana, vol. X (1953), pp. 4152.
[6]Curry, Haskell B. and Feys, Robert, Combinatory logic, vol. 1, North-Holland, Amsterdam, 1958, 1968, 433 pp.
[7]Fraenkel, Abraham A., Der Begriff ‘definit’ und die Unabhängigkeit des Auswahlaxioms, Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse, vol. 21 (1922), pp. 253257; translated in Jean van Heijenoort, From Frege to Gödel, Harvard University Press, Cambridge, 1967, pp. 284–289.
[8]Henkin, Leon, Completeness in the theory of types, this Journal, vol. 15 (1950), pp. 8191; reprinted in [10, pp. 51–63].
[9]Henkin, Leon, A theory of propositional types, Fundamenta Mathematicae, vol. 52 (1963), pp. 323344; errata, ibid., vol. 53 (1963), p. 119.
[10]Hintikka, Jaakko, editor, The philosophy of mathematics, Oxford University Press, Oxford, 1969, 186 pp.
[11]Lévy, Azriel, The Fraenkel-Mostowski method for independence proofs in set theory, The theory of models, Proceedings of the 1963 International Symposium at Berkeley, edited by Addison, J. W., Henkin, Leon, and Tarski, Alfred, North-Holland, Amsterdam, 1965, pp. 221228.
[12]Mostowski, Andrzej, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip, Fundamenta Mathematicae, vol. 32 (1939), pp. 201252.
[13]Sanchis, Luis E., Types in combinatory logic, Notre Dame Journal of Formal Logic, vol. 5 (1964), pp. 161180.
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The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
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