Abstract
Using the method of decisive creatures [see Kellner and Shelah (J Symb Log 74:73–104, 2009)] we show the consistency of “there is no increasing \(\omega _2\)–chain of Borel sets and \(\mathrm{non}({\mathcal N})= \mathrm{non}({\mathcal M})=\mathrm{non}({\mathcal N}\cap {\mathcal M})=\omega _2=2^\omega \)”. Hence, consistently, there are no monotone Borel hulls for the ideal \({\mathcal M}\cap {\mathcal N}\). This answers Balcerzak and Filipczak (Math Log Q 57:186–193, 2011 [Questions 23, 24]). Next we use finite support iteration of ccc forcing notions to show that there may be monotone Borel hulls for the ideals \({\mathcal M},{\mathcal N}\) even if they are not generated by towers.
Notes
Remember our convention that for \(x,y\in {\mathbf H}(i)\) and \({\mathfrak c}\in {\mathbf K}(i)\) we write \(x\in {\mathbf \Sigma }({\mathfrak c})\) iff \(x\in {\mathrm{val}}({\mathfrak c})\), and \(x\in {\mathbf \Sigma }(y)\) iff \(x=y\).
Remember our convention that, for \(x\in {\mathbf H}(i)\), \({\mathrm{val}}(x)=\{x\}\).
“mhg” stands for “monotone hull generating”.
See [16, 3.1–3.7] for the order in which these should be shown.
Since \({\mathbb B}^{\mathbf{V}^{{\mathbb P}^*_{a_i}}}\) is \(\sigma \)–centered we know that the product is ccc.
i.e., determined in a standard way by a sequence of maximal antichains.
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Acknowledgments
Both authors acknowledge support from the United States-Israel Binational Science Foundation (Grant No. 2006108). This is publication 972 of the second author.
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Dedicated to László Fuchs for his ninetieth birthday.
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Rosłanowski, A., Shelah, S. Monotone hulls for \({\mathcal {N}}\cap {\mathcal {M}}\) . Period Math Hung 69, 79–95 (2014). https://doi.org/10.1007/s10998-014-0042-3
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DOI: https://doi.org/10.1007/s10998-014-0042-3