Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T21:10:22.255Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructively accessible ordinal numbers1

Published online by Cambridge University Press:  12 March 2014

Wayne Richter
Wayne Richter*
Affiliation:
Rutgers, The State University
Get access Rights & Permissions [Opens in a new window]

Extract

In [11] the constructive ordinals were extended to constructive finite number classes by using systems of notations where mappings at limit ordinals are just partial recursive functions. It turned out that these systems are equivalent, both in terms of ordinals represented and the forms of the sets of notations, to extensions obtained by using mappings at limit ordinals which are partial recursive in (sets of notations for) previously defined number classes. In this article these results are extended to constructive transfinite number classes. We present a system (F, ||) which, in terms of our analogy with the classical ordinals, provides notations for the ordinals less than the first “constructively inaccessible” ordinal, and show that the above equivalence holds at least this far.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 33 , Issue 1 , 26 April 1968 , pp. 43 - 55
Copyright
Copyright © Association for Symbolic Logic 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

Most of the results presented here were announced without proof in [12]. Research of the author was begun while he held a National Science Foundation Predoctoral fellowship. Some of the material appeared in [10]. The research was continued at Dartmouth College in the summer of 1964, supported by National Science Foundation Grant G23805.

References

[1]Enderton, H. B., Hierarchies in recursive function theory, Transactions of the American Mathematical Society, vol. 111 (1964), pp. 457–471.CrossRefGoogle Scholar
[2]Gandy, R. O., General recursive functionals of finite type and hierarchies of functions, Symposium on mathematical logic, University of Clermont Ferrand, June, 1962 (mimeographed).Google Scholar
[3]Kleene, S. C., Introduction to metamathematics, North-Holland, Amsterdam, Noordhoff, Groningen, van Nostrand, New York and Toronto, 1952.Google Scholar
[4]Kleene, S. C., Recursive functionals and quantifiers of finite types. I, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 1–52.Google Scholar
[5]Kleene, S. C., Recursive functionals and quantifiers of finite types. II, Transactions of the American Mathematical Society, vol. 108 (1963), pp. 106–142.Google Scholar
[6]Kreider, D. L. and Rogers, H. Jr., Constructive versions of ordinal number classes, Transactions of the American Mathematical Society, vol. 100 (1961), pp. 325–369.CrossRefGoogle Scholar
[7]Myhill, J., Creative sets, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 97–108.CrossRefGoogle Scholar
[8]Putnam, H., Uniqueness ordinals in higher constructive number classes, Essays on the foundations of mathematics, Magnes Press, Jerusalem, 1961 and North-Holland, Amsterdam, 1962, pp. 190–206.Google Scholar
[9]Putnam, H., On hierarchies and systems of notations, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 44–50.CrossRefGoogle Scholar
[10]Richter, W., Systems of notations for the “constructively accessible” ordinals, Ph.D. Thesis, Princeton University, 1963.Google Scholar
[11]Richter, W., Extensions of the constructive ordinals, this Journal, vol. 30 (1965), pp. 193–211.Google Scholar
[12]Richter, W., Constructive transfinite number classes, Bulletin of the American Mathematical Society, vol. 73 (1967), pp. 261–265.CrossRefGoogle Scholar
[13]Richter, W., Notation-free hierarchies, in preparation.Google Scholar
[14]Shoenfield, J. R., The form of the negation of a predicate, recursive function theory, Proceedings of the fifth symposia in pure mathematics, American Mathematical Society, Providence, 1962, pp. 131–134.CrossRefGoogle Scholar
[15]Shoenfield, J. R., A hierarchy for objects of type 2, Abstract 65T-173, Notices of the American Mathematical Society, vol. 12 (1965), pp. 369–370.Google Scholar
[16]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955), pp. 151–163.Google Scholar
[17]Tugué, T., Predicates recursive in a type-2 object and Kleene hierarchies, Commentarii Mathematici Universitatis Sancti Pauli, vol. 8 (1960), pp. 97–117.Google Scholar