arXiv subjects
Publications and source records attributed to Andrzej Roslanowski.
We present several remarks on cardinal coefficients associated with the generalized Baire space for an inaccessible cardinal kappa. Our inquiry originates in the work of van der Vlugt arXiv:2307.14118 . We provide evidence that the appropriate coefficients to consider are those determined by binary relations restricted to a club. In other words, we make a case for the systematic use of clubs.
We show that, consistently, there exists a Borel set B subset Cantor admitting a sequence (eta_alpha:alpha 5 for any distinct eta,nu from P. This answers two questions from our previous works.
For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc extension. These sets have uncountably many translations with intersections of size 2i in ZFC, so this answers Problem 3.4 of arxiv:1711.04058 .
We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $α<ω_1$ and $4\leq k<ω$, then there is a ccc forcing notion adding a $Σ^0_2$ set $B\subseteq H$ which has $\aleph_α$ many pairwise $k$--overlapping translations but not a perfect set of such translations.
We give an example of iteration of length omega of (<kappa)-complete kappa^+-cc forcing notions with the limit collapsing kappa^+. The construction is decoded from the proof of Shelah [Proper and Improper Forcing, Appendix, Theorem 3.6(1)].
For a cardinal lambda 5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.
This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+.
We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgrooup of R, while it isconsistent there there is no non-null meager subgroup of R.
Using the method of decisive creatures (math.LO/0601083) we show the consistency of "there is no increasing omega_2 --chain of Borel sets and non(N)=non(M)= omega_2=2^omega". Hence, consistently, there are no monotone hulls for the ideal M cap N . This answers Balcerzak and Filipczak. Next we use FS iteration with partial memory to show that there may be monotone Borel hulls for the ideals M, N even if they are not generated by towers.
We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets in (in R^n). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the sigma-algebra of sets with the property of Baire.
We show the consistency of "there is a nice sigma --ideal I on the reals with add(I)= omega_1 which cannot be represented as the union of a strictly increasing sequence of length omega_1 of sigma-subideals". This answers a question by Borodulin-Nadzieja and Glab.
This article continues Rosłanowski and Shelah math.LO/9906024, math.LO/0508272, math.LO/0210205, math.LO/0611131 and math.LO/0605067. We introduce here a new property of <lambda-strategically complete forcing notions which implies that their lambda-support iterations do not collapse lambda^+ (for a strongly inaccessible cardinal lambda).
We show that there exist two proper creature forcings having a simple (Borel) definition, whose product is not proper. We also give a new condition ensuring properness of some forcings with norms.
We show a general scheme of Ramsey-type results for partitions of countable sets of finite functions, where "one piece is big" is interpreted in the language originating in creature forcing. The heart of our proofs follows Glazer's proof of the Hindman Theorem, so we prove the existence of idempotent ultrafilters with respect to suitable operation. Then we deduce partition theorems related to creature forcings.
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in math.LO/0407498. We introduce stronger properties of ultrafilters and we show that those properties may be handled in lambda-support iterations of reasonably bounding forcing notions. We use this to show that consistently there are reasonable ultrafilters on an inaccessible cardinal lambda with generating system of size less than 2^lambda . We also show how reasonable ultrafilters can be killed by forcing notions which have enough reasonable completeness to be iterated with lambda-supports (and we show the appropriate preservation theorem).
We introduce several properties of forcing notions which imply that their lambda-support iterations are lambda-proper. Our methods and techniques refine those studied in math.LO/9906024, math.LO/0210205, math.LO/0508272 and math.LO/0605067, covering some new forcing notions (though the exact relation of the new properties to the old ones remains undecided).
We continue investigations of reasonable ultrafilters on uncountable cardinals defined in Shelah math.LO/0407498 and studied also in math.LO/0605067. We introduce a general scheme of generating a filter on lambda from filters on smaller sets and we investigate the combinatorics of objects obtained this way.