Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T23:43:04.148Z Has data issue: false hasContentIssue false
  • English
  • Français

Measure theory aspects of locally countable orderings

Published online by Cambridge University Press:  12 March 2014

Liang Yu
Liang Yu*
Affiliation:
Department of Mathematics, Faculty of Science, National University of Singapore, Lower Kent Ridge Road, Singapore 117543, Singapore.E-mail:[email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that for any locally countable partial order ℙ = (2ε, ≤p, there exists a nonmeasurable antichain in ℙ. Some applications of the result are also presented.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 3 , September 2006 , pp. 958 - 968
Copyright
Copyright © Association for Symbolic Logic 2006

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]Billingsley, Patrick, Probability and measure, third ed., Wiley Series in Probability and Mathematical Statistics, A Wiley-Interscience Publication, John Wiley Sons, Inc., New York, 1995.Google Scholar
[2]Downey, R. and Hirschfeldt, D., Algorithmic randomness and complexity, Monographs in Computer Science, Springer-Verlag, in preparation.Google Scholar
[3]Friedman, H., Borel structures in mathematics, manuscript, Ohio State Univeristy, 1979.Google Scholar
[4]Harrington, L., Marker, D., and Shelah, S., Borel orderings, Transactions of the American Mathematical Society, vol. 310 (1988), no. 1, pp. 293–302.CrossRefGoogle Scholar
[5]Harrison, J., Some applications of recursive pseudo-well-orderings, Ph.D. thesis, Stanford University, 1966.Google Scholar
[6]Jech, T., Set theory, The third millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.Google Scholar
[7]Kautz, S., Degrees of random sets, Ph.D. thesis, Cornell, 1991.Google Scholar
[8]Kurtz, S., Randomness and genericity in the degrees of unsolvability, Ph.D. thesis, University of Illinois, Urbana, 1981.Google Scholar
[9]Lerman, M., Degrees of unsolvability. Local and global theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1983.CrossRefGoogle Scholar
[10]Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[11]Miller, J., The K-degrees, low for K degrees, and weakly low for K oracles, to appear.Google Scholar
[12]Miller, J. and Yu, Liang, On initial segment complexity and degrees of randomness, to appear in Transactions of the American Mathematical Society.Google Scholar
[13]Moschovakis, Y., Descriptive set theory, Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland Publishing Co., Amsterdam, New York, 1980.Google Scholar
[14]Sacks, G., Degrees of unsolvability, Princeton University Press, Princeton, N.J., 1963.Google Scholar
[15]Sacks, G., Measure-theoretic uniformity in recursion theory and set theory, Transactions of the American Mathematical Society, vol. 142 (1969), pp. 381–420.CrossRefGoogle Scholar
[16]Sacks, G., Higher recursion theory, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1990.CrossRefGoogle Scholar
[17]Lambalgen, M. Van, Random sequences, Ph.D. thesis, University of Amsterdam, 1987.Google Scholar