Abstract
My main purpose here is to describe an abstract, algebraic context in which one can develop at least the rudiments of the theory of inductive definability. It will be obvious that all the standard examples of induction fit the abstract model in a natural way: these include ordinary recursion theory on the integers and its generalizations to various kinds of abstract structures as well as the theories of positive and non-monotone elementary induction, e.g. see Moschovakis (1974a, b). Despite this generality, the model is very simple and one can establish for it easily the basic results which are common to all these theories. My hope is that future monographs on inductive definability will start with a brief chapter on ‘algebraic preliminaries’ which will contain some version of this material.
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Bibliography
Aczel, P. and Richter, W.: 1972, ‘Inductive Definitions and Analogues of Large Cardinals’, Conf. in Mathematical Logic, London 1970
Aczel, P. and Richter, W.: 1972, Springer Lecture Notes in Math. 255, 1–9.
Aczel, P. and Richter, W.: 1974 ‘Inductive Definitions and Reflecting Properties of Admissible Ordinals’, in J. E. Fenstad and P. Hinman (eds.), Generalized Recursion Theory, North Holland, Amsterdam.
Gandy, R. O.: 1962, ‘General Recursive Functionals of Finite Type and Hierarchies of Functions’, in Proc. Logic Colloq. Clermont Ferrand, pp. 5-24.
Grilliot, T. J.: 1969, ‘Selection Functions for Recursive Functionals’, Notre Dame Journal of Formal Logic 10, 225–234.
Grilliot, T. J.: 1971, ‘Inductive Definitions and Computability’, Trans. Am. Math. Soc. 158, 309–317.
Hinman, P. G.: 1969, ‘Hierarchies of Effective Descriptive Set Theory’, Trans. Am. Math. Soc. 142, 111–140.
Kechris, A. S. and Moschovakis, Y. N.: a, ‘Recursion in Higher Types’, to appear in K. J. Barwise (ed.), Handbook of Mathematical Logic..
Moschovakis, Y. N.: 1967, ‘Hyperanalytic Predicates’, Trans. Am. Math. Soc. 129, 249–282.
Moschovakis, Y. N.: 1969, ‘Abstract First Order Computability I and if’, Trans. Am. Math. Soc. 138, 427–504.
Moschovakis, Y. N.: 1974a, Elementary Induction on Abstract Structures, North Holland, Amsterdam, 1974.
Moschovakis, Y. N.: 1974b, ‘On Nonmonotone Inductive Definability’, Fund. Math. 82, 39–83.
Platek, R.: 1966, ‘Foundations of Recursion Theory’, Doctoral Dissertation, Stanford University, Stanford, California.
Richter, W: 1971, ‘Recursively Mahlo Ordinals and Inductive Definitions’, in: R. O. Gandy and C. E. M. Yates (eds.), Logic Colloquium’ 69, North Holland, Amsterdam.
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© 1977 D. Reidel Publishing Company, Dordrecht, Holland
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Moschovakis, Y.N. (1977). On the Basic Notions in the Theory of Induction. In: Butts, R.E., Hintikka, J. (eds) Logic, Foundations of Mathematics, and Computability Theory. The University of Western Ontario Series in Philosophy of Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1138-9_12
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DOI: https://doi.org/10.1007/978-94-010-1138-9_12
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