arXiv subjects
Publications and source records attributed to Andrzej Rosłanowski.
We develop the theory of the forcing with trees and creatures for an inaccessible lambda continuing Rosłanowski and Shelah math.LO/9807172, math.LO/9909115. To make a real use of these forcing notions (that is to iterate them without collapsing cardinals) we need suitable iteration theorems, and those are proved as well. (In this aspect we continue Rosłanowski and Shelah math.LO/9906024.)
We show that the reduced cofinality of the nonstationary ideal NS_kappa on a regular uncountable cardinal kappa may be less than its cofinality, where the reduced cofinality of NS_kappa is the least cardinality of any family F of nonstationary subsets of kappa such that every nonstationary subset of kappa can be covered by less than $κ$ many members of F.
Assume V=L and lambda is regular smaller than the first weakly compact cardinal. Under those circumstances and with arbitrary requirements on the structure of Ext(G,Z) (under well known limitations), we construct an abelian group G of cardinality lambda such that for no G' subseteq G, |G'|< lambda is G/G' free and Ext(G,Z) realizes our requirements.
We address ZFC inequalities between some cardinal invariants of the continuum, which turned to be true in spite of strong expectations given by [RoSh:470].
We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.
We use the method of norms on possibilities to answer a question of Kunen and construct a ccc sigma --ideal on 2^omega with various closure properties and distinct from the ideal of null sets, the ideal of meager sets and their intersection.
Let I be an ideal of subsets of a Polish space X, containing all singletons and possessing a Borel basis. Assuming that I does not satisfy ccc, we consider the following conditions (B), (M) and (D). Condition (B) states that there is a disjoint family F subseteq P(X) of size c, consisting of Borel sets which are not in I. Condition (M) states that there is a function f:X-> X with f^{-1}[{x}] notin I for each x in X. Provided that X is a group and I is invariant, condition (D) states that there exist a Borel set B notin I and a perfect set P subseteq X for which the family {B+x: x in P} is disjoint. The aim of the paper is to study whether the reverse implications in the chain (D) => (M) => (B) => not-ccc can hold. We build a sigma-ideal on the Cantor group witnessing''(M) and not (D)'' (Section 2). A modified version of that sigma-ideal contains the whole space (Section 3). Some consistency results deriving (M) from (B) for''nicely'' defined ideals are established (Section 4). We show that both ccc and (M) can fail (Theorems 1.3 and 4.2). Finally, some sharp versions of (M) for invariant ideals on Polish groups are investigated (Section 5).
In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [JR1] in which several problems were posed. We answer some of those problems here.
In the present paper we are interested in properties of forcing notions which measure in a sense the distance between the ground model reals and the reals in the extension. We look at the ways the ``new'' reals can be aproximated by ``old'' reals. We consider localizations for infinite subsets of omega. Though each member of [omega]^omega can be identified with its increasing enumeration, the (standard) localizations of the enumeration does not provide satisfactory information on successive points of the set. They give us ``candidates'' for the n-th point of the set but the same candidates can appear several times for distinct n. That led to a suggestion that we should consider disjoint subsets of omega as sets of ``candidates'' for successive points of the localized set. We have two possibilities. Either we can demand that each set from the localization contains a limited number of members of the localized set or we can postulate that each intersection of that kind is large. Localizations of this kind are studied in section 1. In the second section we investigate localizations of infinite subsets of omega by sets of integers from the ground model. These localizations might be thought as localizations by partitions of omega into successive intervals.
We prove that the Sacks forcing collapses the continuum onto the dominating number d, answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses omega_2 then it forces diamond_{omega_1} .
We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions maybe large while the covering for measure is small.
We give a model where there is a ccc Souslin forcing which does not satisfy the Knaster condition. Next, we present a model where there is a sigma-linked not sigma-centered Souslin forcing such that all its small subsets are sigma-centered but Martin Axiom fails for this order. Furthermore, we construct a totally nonhomogeneous Souslin forcing and we build a Souslin forcing which is proper but not ccc that does not contain a perfect set of mutually incompatible conditions. Finally we show that ccc Sigma^1_2-notions of forcing may not be indestructible ccc.