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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.

MSC:

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

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