Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T06:35:21.755Z Has data issue: false hasContentIssue false
  • English
  • Français

The existence of countable totally nonconstructive extensions of the countable atomless Boolean algebra

Published online by Cambridge University Press:  12 March 2014

E. W. Madison
E. W. Madison*
Affiliation:
University of Iowa, Iowa City, Iowa 52242
Get access Rights & Permissions [Opens in a new window]

Abstract

Our results concern the existence of a countable extension of the countable atomless Boolean algebra such that is a “nonconstructive” extension of . It is known that for any fixed admissible indexing φ of there is a countable nonconstructive extension of (relative to φ). The main theorem here shows that there exists an extension of such that for any admissible indexing φ of , is nonconstructive (relative to φ).Thus, in this sense a countable totally nonconstructive extension of .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 1 , March 1983 , pp. 167 - 170
Copyright
Copyright © Association for Symbolic Logic 1983

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.)

References

REFERENCES

[1]Alton, D.A. and Madison, E.W., Computability of Boolean algebras and their extensions, Annals of Mathematical Logic, vol. 6 (1973), pp. 95–128.CrossRefGoogle Scholar
[2]Halmos, P.R., Lectures on Boolean algebras, Van Nostrand, Princeton, 1963.Google Scholar
[3]Madison, E.W., A hierarchy of regular open sets of the Cantor space, Acta Mathematica (to appear).Google Scholar
[4]Madison, E.W. and Nelson, G.C., Some examples of constructive and nonconstructive extensions of the countable atomless Boolean algebra, Journal of the London Mathematical Society (2), vol. 11 (1975), pp. 325–336.CrossRefGoogle Scholar
[5]Rabin, M.O., Computable algebras, Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
[6]Rogers, H., Theory of recursive functions and effective computability, McGraw-Hall, New York, 1967.Google Scholar
[7]Sikorski, Roman, Boolean algebras, 2nd edition, Academic Press, New York, 1964.Google Scholar