Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-28T21:57:52.435Z Has data issue: false hasContentIssue false
  • English
  • Français

Definability of measures and ultrafilters

Published online by Cambridge University Press:  12 March 2014

David Pincus  and
Robert M. Solovay
David Pincus
Affiliation:
University of Washington, Seattle, Washington 98195 Thomas J. Watson Research Center, Yorktown Heights, New York 10598
Robert M. Solovay
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 Thomas J. Watson Research Center, Yorktown Heights, New York 10598
Get access Rights & Permissions [Opens in a new window]

Extract

Nonprincipal ultrafilters are harder to define in ZFC, and harder to obtain in ZF + DC, than nonprincipal measures.

The function μ from P(X) to the closed interval [0, 1] is a measure on X if μ. is finitely additive on disjoint sets and μ(X) = 1. (P is the power set.) μ is nonprincipal if μ ({x}) = 0 for each x Є X. μ is an ultrafilter if Range μ= {0, 1}. The existence of nonprincipal measures and ultrafilters on any infinite X follows from the axiom of choice.

Nonprincipal measures cannot necessarily be defined in ZFC. (ZF is Zermelo–Fraenkel set theory. ZFC is ZF with choice.) In ZF alone they cannot even be proved to exist. This was first established by Solovay [14] using an inaccessible cardinal. In the model of [14] no nonprincipal measure on ω is even ODR (definable from ordinal and real parameters). The HODR (hereditarily ODR) sets of this model form a model of ZF + DC (dependent choice) in which no nonprincipal measure on ω exists. Pincus [8] gave a model with the same properties making no use of an inaccessible. (This model was also known to Solovay.) The second model can be combined with ideas of A. Blass [1] to give a model of ZF + DC in which no nonprincipal measures exist on any set. Using this model one obtains a model of ZFC in which no nonprincipal measure on the set of real numbers is ODR. H. Friedman, in private communication, previously obtained such a model of ZFC by a different method. Our construction will be sketched in 4.1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 2 , June 1977 , pp. 179 - 190
Copyright
Copyright © Association for Symbolic Logic 1977

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

BIBLIOGRAPHY

[1]Blass, A., A model without ultrafilters (to appear).Google Scholar
[2]Feferman, S., Some applications of the notions of forcing and generic sets, Fundamenta Mathematicae, vol. 56 (1965), pp. 325–345.Google Scholar
[3]Halpern, J. D. and Levy, A., The Boolean prime ideal theorem does not imply the axiom of choice, Proceedings of Symposia in Pure Mathematics, XIII, Part I, pp. 83–134.Google Scholar
[4]Levy, A., On the logical complexity of several axioms of set theory, Proceedings of Symposia in Pure Mathematics, XIII, Part I, pp. 219–230.Google Scholar
[5]Luxemburg, W. A. J., Reduced products of the real number system, Applications of model theory to algebra, analysis and probability, Holt Reinbout and Winston, New York, 1969.Google Scholar
[6]Maharam, D., On homogeneous measure algebras, Proceedings of the National Academy of Sciences, vol. 28 (1942), pp. 108–111.CrossRefGoogle ScholarPubMed
[7]McAloon, K., Consistency results about ordinal definability, Annals of Mathematical Logic, vol. 2 (1971), pp. 449–467.Google Scholar
[8]Pincus, D., The strength of the Hahn–Banach Theorem, Victoria Symposium on Nonstandard Analysis, Springer Lecture Notes, No. 369, 1974, pp. 203–248.CrossRefGoogle Scholar
[9]Pincus, D., Adding dependent choice, Annals of Mathematical Logic (to appear).Google Scholar
[10]Royden, H., Real analysis, Macmillan, New York, 1968.Google Scholar
[11]Rudin, W., Real and complex analysis, McGraw-Hill, New York, 1966.Google Scholar
[12]Sacks, G. E., Measure theoretic uniformity in recursion theory and axiomatic set theory, Transactions of the American Mathematical Society, vol. 142 (1969), pp. 381–420.CrossRefGoogle Scholar
[13]Sierpinski, W., Fonctions additives non-complètement additives et fonctions non-measurables, Fundamenta Mathematicae, vol. 30 (1938), pp. 96–99.Google Scholar
[14]Solovay, R. M., A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 1–56.Google Scholar
[15]Solovay, R. M., Real valued measurable cardinals, Proceedings of Symposia in Pure Mathematics, XIII, Part I, pp. 397–428.Google Scholar