Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-26T14:16:09.622Z Has data issue: false hasContentIssue false
  • English
  • Français

Rich co-ordinals, addition isomorphisms, and RETs1

Published online by Cambridge University Press:  12 March 2014

Alfred B. Manaster
Alfred B. Manaster*
Affiliation:
Massachusetts Institute of Technology and Institute for Defense Analyses
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper a special type of co-ordinal, called rich, is studied. Basic properties of rich co-ordinals are proved in §1. In §2 rich co-ordinals are seen to be the co-ordinals occurring in paths which are addition isomorphic to initial segments of the classical ordinals. The results of §1 are applied to obtain information about and examples of such paths. In the next section the order types of rich co-ordinals with a given field, X, is seen to be determined essentially by the finite divisors of X. RETs satisfying various divisibility conditions are constructed in §4.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 1 , 29 May 1969 , pp. 45 - 52
Copyright
Copyright © Association for Symbolic Logic 1969

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

Part of the research reported here was supported under NSF grant GP 6982.

References

[1]Crossley, J. N., Constructive order types. I, Formal systems and recursive functions, Proceedings of the Eighth Logic Colloquium, Oxford 1963, Ed. by Crossley, J. N. and Dummett, M. A. E., Amsterdam, 1965, pp. 189–264.CrossRefGoogle Scholar
[2]Crossley, J. N., Constructive order types. II, this Journal, vol. 31 (1966), pp. 525–538.Google Scholar
[3]Dekker, J. C. E., The factorial function for isols, Mathematische Zeitschrift, vol. 70 (1958), pp. 250–262.CrossRefGoogle Scholar
[4]Dekker, J. C. E. and Myhill, J., Recursive equivalence types, University of California Publications in Mathematics (New Series) vol. 3, No. 3 (1960), 67–214.Google Scholar
[5]Friedberg, R. M., The uniqueness of finite division for recursive equivalence types, Mathematische Zeitschrift, vol. 75 (1960/1961), pp. 3–7.CrossRefGoogle Scholar
[6]Manaster, A. B., Higher-order indecomposable isols, Ph.D. thesis, Cornell University, 1965.Google Scholar
[7]Manaster, A. B., Full co-ordinals of RETs, Pacific Journal of Mathematics (to appear).Google Scholar
[8]Nerode, A., Additive relations among recursive equivalence types, Mathematische Annalen, vol. 159 (1965), pp. 329–343.CrossRefGoogle Scholar
[9]Sierpinski, W., Cardinal and ordinal numbers, Polska Akademia Nauk, Warszawa, 1958.Google Scholar