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Dominical categories: recursion theory without elements1 2

Published online by Cambridge University Press:  12 March 2014

Robert A. di Paola  and
Alex Heller
Robert A. di Paola
Affiliation:
Department of Computer Science, Queens College, City University of New York, Flushing, New York 11367 Department of Mathematics, Graduate School and University Center, City University of New York, New York, New York 10036
Alex Heller
Affiliation:
Department of Mathematics, Graduate School and University Center, City University of New York, New York, New York 10036
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Extract

Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem.

Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developed without reference to elements.

By Gödel's generalized incompleteness theorem for consistent arithmetical system T we mean any statement of the following sort:

(1) if every recursive set is definable in T, then T is essentially undecidable [41]; or

(2) if all recursive functions are definable in T, then T is essentially undecidable [41]; or

(3) if every recursive set is definable in T, then T 0 and R 0 (the sets of Gödel numbers of the theorems and refutables of T) are recursively inseparable [39]; or

(4) if all re sets are representable in T, then T 0 is creative [28], [39]; or

(5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T), then T 0 and R 0 are effectively inseparable [39].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 52 , Issue 3 , September 1987 , pp. 594 - 635
Copyright
Copyright © Association for Symbolic Logic 1987

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Footnotes

1

Some of the contents of this paper were presented by both authors at the Fifth and Sixth Congresses of the School of Specialization in Mathematical Logic of the University of Siena, held at Siena, Italy in April 1983 and January 1984, respectively. The research leading to the paper was sponsored by the National Science Foundation Cooperative Science Program. Preparation of the paper was also supported by grant PSC-BHE 6-62054 of the City University of New York to the first named author, who would like to acknowledge also early support on this general subject from the Penrose Fund of the American Philosophical Society.

2

This paper formed the basis of an invited address by the first named author at the Annual Meeting of the Association of Symbolic Logic, held in conjunction with the American Philosophical Association in Washington, D.C., December 28–30 1985.

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