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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

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