Defining general structures.
Jack C. Boudreaux · Notre Dame Journal of Formal Logic · 1979
0 Introduction Henkin demonstrated in [7] that an adequate semantic theory for any axiomatic higher-order functional calculi could be developed, if the class of structures (viz., models, interpretations, realizations, etc.) upon which the semantic theory is based is significantly wider than the class of all standard structures. Moreover, if the calculus contains the axiom schemas of Extensionality and Comprehension, then it is obvious that the members of the appropriate wider class of structures, which may be called the class of general structures, must exhibit a rather high degree of internal organization. In [9], p. 324, Henkin observed that an important technical problem had remained unsolved, i.e., to give a perspicuous definition of this class of structures which is not overly dependent upon the syntactic design of the higher-order language. Andrews proposed one solution to this problem in [1] and [2]; he proved that every general structure must be closed with respect to certain combinatory operators. In this paper I will propose an alternative solution in which the definition of the class of general structures is given in strictly set-theoretical terms. Specifically, I will prove that every general structure must be closed with respect to a family of Projectiυe operations, cf. Kuratowski and Mostowski [10], pp. 357-358, and a Cut operation, which I have adapted from Shoenfield [12], p. 230.