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Primitive recursion, equality, and a universal set

Published online by Cambridge University Press:  04 March 2009

M. Pfender ,
M. Kröplin  and
D. Pape
M. Pfender
Affiliation:
Technische Universität Berlin, MA 7-3, Str. d. 17. Juni 136, D-10623 Berlin
M. Kröplin
Affiliation:
Technische Universität Berlin, MA 7-3, Str. d. 17. Juni 136, D-10623 Berlin
D. Pape
Affiliation:
Technische Universität Berlin, MA 7-3, Str. d. 17. Juni 136, D-10623 Berlin
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Abstract

Within a categorical framework for primitive recursion, equality between p.r. maps is shown to be definable by suitable p.r. equality predicates. Equivalence is shown between a direct categorical formalization of classical p.r. functions and p.r. maps in the sense of Lawvere and Freyd. An extension of the theory is shown to admit a ‘universal set’ containing all objects of the extended theory of subobjects.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 4 , Issue 3 , September 1994 , pp. 295 - 313
Copyright
Copyright © Cambridge University Press 1994

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References

Davis, M. (1958) Computability and unsolvability, McGraw-Hill.Google Scholar
Eilenberg, S. and Elgot, C. C. (1970) Recursiveness, Academic Press.Google Scholar
Freyd, P. (1972) Aspects of topoi. Bull. Australian Math. Soc. 7 1–76.CrossRefGoogle Scholar
Goodstein, R. L. (1971) Development of mathematical logic, Logos Press.Google Scholar
Lambek, J. and Scott, P. J. (1986) Introduction to higher order categorical logic, Cambridge University Press.Google Scholar
Lawvere, F. W. (1963) Functorial semantics of algebraic theories. Proc. Nat. Acad. Sci. USA 50 869–872.CrossRefGoogle ScholarPubMed
Lawvere, F. W. (1964) An elementary theory of the category of sets. Proc. Nat. Acad. Sci. USA 51 1506–1510.CrossRefGoogle Scholar
Pape, D. (1989) Ein vereinfachtes Schema für primitive Rekursion und ein darauf aufbauender Beweis, daβ die Ackermann-Funktion nicht primitiv rekursiv ist. Vortragsausarbeitung Seminar ‘Rekursionstheorie’, E. J. Thiele TU Berlin.Google Scholar
Péter, R. (1967) Recursive functions (3rd ed.), Academic Press.Google Scholar
Pfender, M., Reiter, R. and Sartorius, M. (1982) Constructive arithmetics. Lecture Notes in Mathematics 962 228–236.CrossRefGoogle Scholar
Reiter, R. (1980) Mengentheoretische Konstruktionen in arithmetischen Universen, Diploma thesis, TU Berlin.Google Scholar
Reiter, R. (1982) Ein algebraisch-konstruktives Abbildungskalkül zur Fundierung der elementaren Arithmetik, Dissertation, Berlin (rejected by TUB).Google Scholar
Román, L. (1989) Cartesian categories with natural numbers object. J. Pure Appl. Algebra 58 267–278.CrossRefGoogle Scholar
Skolem, , Th. (1923) Begründung der elementären Arithmetik durch die rekurrierende Denkweise ohne Anwendung scheinbarer Veränderlichen mit unendlichem Ausdehnungsbereich. K. V. Skr. I 6 38 ff. (Reprinted in: Fenstadt, J. E. (ed.) (1970) Th. Skolem. Selected works in logic, Oslo: Universitetsforlaget.)Google Scholar
Tarski, A. and Givant, S. (1987) A formalization of set theory without variables, American Mathematical Society.CrossRefGoogle Scholar
Yashuhara, A. (1971) Recursive function theory and logic, Academic Press.Google Scholar