Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-17T03:54:46.955Z Has data issue: false hasContentIssue false

Strong measure zero sets without Cohen reals

Published online by Cambridge University Press:  12 March 2014

Martin Goldstern*
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel, E-mail: judah@bimacs.cs.biu.ac.il
Haim Judah
Affiliation:
Department of Mathematics, Bar Ilan University, 52900 Ramat Gan, Israel, E-mail: judah@bimacs.cs.biu.ac.il
Saharon Shelah
Affiliation:
Department of Mathematics, Givat Ram, Hebrew University of Jerusalem, 91904 Jerusalem, Israel, E-mail: shelah@math. huji.ac.il
*
2. Mathematisches Institut, Freie Universität Berlin, 14195 Berlin, Germany, E-mail: goldstrn@math.fu-berlin.de

Abstract

If ZFC is consistent, then each of the following is consistent with :

(1) X ⊆ ℝ is of strong measure zero iff ∣X∣ ≤ ℵ1 + there is a generalized Sierpinski set.

(2) The union of ℵ many strong measure zero sets is a strong measure zero set + there is a strong measure zero set of size ℵ2 + there is no Cohen real over L.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1993

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]Abraham, U. and Shelah, S., Isomorphism types of Aronszajn trees, Israel Journal of Mathematics, vol. 50 (1985), pp. 75113.CrossRefGoogle Scholar
[2]Baumgartner, J., Iterated forcing, Surveys in set theory (Mathias, A. R. D., editor), London Mathematical Society Lecture Note Series, No. 8, Cambridge University Press, Cambridge, 1983.Google Scholar
[3]Blass, A. and Shelah, S., There may be simple and -points, and the Rudin-Keisler order may be downward directed, Annals of Pure and Applied Logic, vol. 33 (1987), pp. 213243.CrossRefGoogle Scholar
[4]Carlson, T., Strong measure zero and strongly meager sets, Proceedings of the American Mathematical Society, vol. 118 (1993), pp. 577586.CrossRefGoogle Scholar
[5]Corazza, P., The generalized Borel conjecture and strongly proper orders, Transactions of the American Mathematical Society, vol. 316 (1989), pp. 115140.CrossRefGoogle Scholar
[6]Goldstern, M., Tools for your forcing construction, Proceedings of the 1991 Bar Ilan conference on Set Theory of the Reals (Judah, H., editor), Israel Mathematical Conference Proceedings, vol. 6, 1993.Google Scholar
[7]Groszek, M. and Jech, T., Generalized iteration of forcing, Transactions of the American Mathematical Society, vol. 324 (1991), pp. 126.CrossRefGoogle Scholar
[8]Judah, H., Shelah, S., and Woodin, H., The Borel conjecture, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 255269.CrossRefGoogle Scholar
[9]Judah, H. and Shelah, S., MA(ω-centered): Cohen reals, strong measure zero sets and strongly meager sets, Israel Journal of Mathematics, vol. 68 (1989), pp. 117.CrossRefGoogle Scholar
[10]Judah, H., Strong measure zero sets and rapid filters, this Journal, vol. 53 (1988), pp. 393402.Google Scholar
[11]Kunen, K., Set theory: An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North-Holland, Amsterdam, 1980.Google Scholar
[12]Miller, A., Mapping a set of reals onto the reals, this Journal, vol. 48 (1983), pp. 575584.Google Scholar
[13]Miller, A., Rational perfect set forcing, Axiomatic set theory, Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, Rhode Island, 1984, pp. 143159.CrossRefGoogle Scholar
[14]Miller, A., Some properties of measure and category, Transactions of the American Mathematical Society, vol. 266 (1981), pp. 93114.CrossRefGoogle Scholar
[15]Pawlikowski, J., Power of transitive bases of measure and category, Proceedings of the American Mathematical Society, vol. 93 (1985), pp. 719729.CrossRefGoogle Scholar
[16]Pawlikowski, J., Finite support iteration and strong measure zero sets, this Journal, vol. 55 (1990), pp. 674677.Google Scholar
[17]Rothberger, F., Sur des families indenombrables de suites de nombres naturels et les problèmes concernant la proprieté C, Proceedings of the Cambridge Philosophical Society, vol. 37 (1941), pp. 109126.Google Scholar
[18]Rothberger, F., Eine Verschärfung der Eigenschaft C, Fundamenta Mathematicae, vol. 30 (1938), pp. 5055.CrossRefGoogle Scholar
[19]Shelah, S., Proper forcing, Lecture Notes in Mathematics, vol. 942, Springer-Verlag, Berlin and New York, 1982.CrossRefGoogle Scholar
[20]Shelah, S., Proper and improper forcing, Perspectives in Mathematics, Springer-Verlag.Google Scholar
[21]Shelah, S., Some notes on iterated forcing with , Notre Dame Journal of Formal Logic, vol. 29 (1988), pp. 117.Google Scholar
[22]Solovay, R. and Tennenbaum, S., Iterated Cohen extensions and Souslin's problem, Annals of Mathematics, vol. 94 (1971), pp. 201245.CrossRefGoogle Scholar
[23]Solovay, R., Real valued measurable cardinals, Axiomatic set theory (Scott, D., editor), Proceedings of Symposia in Pure Mathematics, vol. 13, Part 1, American Mathematical Society, Providence, Rhode Island, 1971, pp. 397428.CrossRefGoogle Scholar
[24]Veličkovič, B., CCC posets of perfect trees, preprint, Compositio Mathematica, vol. 79 (1991), pp. 279294.Google Scholar