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Some properties of the syntactic p-recursion categories generated by consistent, recursively enumerable extensions of Peano arithmetic.

Published online by Cambridge University Press:  12 March 2014

Robert A. Di Paola  and
Franco Montagna
Robert A. Di Paola
Affiliation:
Queens College and the Graduate Center, City University of New York, New York, New York 10036
Franco Montagna
Affiliation:
Dipartimento di Matematica, Universita di Siena, 53100 Siena, Italy
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Extract

The dominical categories were introduced by Di Paola and Heller, as a first step toward a category-theoretic treatment of the generalized first Godel incompleteness theorem [1]. In his Ph.D. dissertation [7], Rosolini subsequently defined the closely related p-categories, which should prove pertinent to category-theoretic representations of incompleteness for intuitionistic systems. The precise relationship between these two concepts is as follows: every dominical category is a pointed p-category, but there are p-categories, indeed pointed p-isotypes (all pairs of objects being isomorphic) with a Turing morphism that are not dominical. The first of these assertions is an easy consequence of the fact that in a dominical category C by definition the near product functor when restricted to the subcategory Ct, of total morphisms of C (as “total” is defined in [1]) constitutes a bona fide product such that the derived associativity and commutativity isomorphisms are natural on C × C × C and C × C, respectively, as noted in [7]. As to the second, p-recursion categories (that is, pointed p-isotypes having a Turing morphism) that are not dominical were defined and studied by Montagna in [6], the syntactic p-categories ST and S′T associated with consistent, recursively enumerable extensions of Peano arithmetic, PA. These merit detailed investigation on several counts.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 2 , June 1991 , pp. 643 - 660
Copyright
Copyright © Association for Symbolic Logic 1991

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References

REFERENCES

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