Andrzej Rosłanowski

COFINALITY OF THE NONSTATIONARY IDEAL

We show that the reduced cofinality of the nonstationary ideal AS κ on a regular uncountable cardinal κ may be less than its cofinality, where the reduced cofinality of NS κ is the least cardinality of any family F of nonstationary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F. For this we investigate connections of the various cofinalities of NS κ with other cardinal characteristics of κ κ and we also give a property of forcing notions (called manageability) which is preserved in <κ-support iterations and which implies that the forcing notion preserves non-meagerness of subsets of κ κ (and does not collapse cardinals nor changes cofinalities).

Sweet & sour and other flavours of ccc forcing notions

Abstract.We continue developing the general theory of forcing notions built with the use of norms on possibilities, this time concentrating on ccc forcing notions and classifying them.

More about λ-support iterations of (<λ)-complete forcing notions

This article continues Rosłanowski and Shelah (Int J Math Math Sci 28:63–82, 2001; Quaderni di Matematica 17:195–239, 2006; Israel J Math 159:109–174, 2007; 2011; Notre Dame J Formal Logic 52:113–147, 2011) and we introduce here a new property of (<λ)-strategically complete forcing notions which implies that their λ-support iterations do not collapse λ+ (for a strongly inaccessible cardinal λ).

Partition Theorems from Creatures and Idempotent Ultrafilters

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.

Universal forcing notions and ideals

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters.

Monotone hulls for {\mathcal {N}}\cap {\mathcal {M}}

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

SHEVA-SHEVA-SHEVA: LARGE CREATURES

omega _2

FORCING FOR hL AND hd

ω2–chain of Borel sets and