Dual Borel conjecture and Cohen reals. (English) Zbl 1213.03063

The Borel Conjecture (BC) states that every strong measure zero set is countable and the Dual Borel Conjecture (DBC) states that every strong meager set is countable. The paper was motivated by the question whether \(\text{BC}+\text{DBC}\) is consistent with ZFC.
It is known that BC implies \(\text{cov}(\mathcal M)=\omega_1\) and DBC implies \(\text{cov}(\mathcal N)=\omega_1\), where \(\mathcal M\) and \(\mathcal N\) denote, respectively, the \(\sigma\)-ideal of meager sets and the \(\sigma\)-ideal of null sets. All known models for DBC satisfy \(\text{cov}(\mathcal N)>\omega_1\).
In the paper under review, the authors prove that \(\text{DBC}+\text{cov}(\mathcal M)=\omega_1\) is consistent with ZFC. Moreover, \(\text{non}(\mathcal N)=\omega_1\) in the presented model.


03E35 Consistency and independence results
03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
Full Text: DOI


[1] DOI: 10.1007/BF02764965 · Zbl 0693.03032
[2] DOI: 10.1090/S0002-9939-1993-1139474-6
[3] Set Theory: on the structure of the real line (1995)
[4] Proper and improper forcing (1998) · Zbl 0889.03041
[5] DOI: 10.1016/0168-0072(90)90058-A · Zbl 0717.03020
[6] Norms on possibilities I: forcing with trees and creatures (1999) · Zbl 0940.03059
[7] Finite support iteration and strong measure zero sets 55 pp 674– (1990)
[8] DOI: 10.1090/S0002-9939-1954-0063389-3
[9] DOI: 10.1007/BF02392416 · Zbl 0357.28003
[10] DOI: 10.1515/JAA.2006.1 · Zbl 1113.03044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.