Factorials of infinite cardinals
Families of almost disjoint Hamel bases.
For infinite dimensional Banach spaces X we investigate the maximal size of a family of pairwise almost disjoint normalized Hamel bases of X, where two sets A and B are said to be almost disjoint if the cardinality of A ∩ B is smaller than the cardinality of either A or B.
Fat P-sets in the Space ω*
We prove that-consistently-in the space ω* there are no P-sets with the ℂ-cc and any two fat P-sets with the ℂ⁺-cc are coabsolute.
Filter games on ω and the dual ideal
We continue the efforts to characterize winning strategies in various infinite games involving filters on the natural numbers in terms of combinatorial or structural properties of the given filter. Previous results in the literature included those games where player II responded with natural numbers, or finite subsets of natural numbers. In this paper we concentrate on games where player II responds with members of the dual ideal. We also give a summary of known results on filter games.
Finiteness and choice
We deal with weak choice principles of the form: Every "finite" family of non-empty sets has a choice function, where "finite" stands for one of several different definitions of finiteness that are not equivalent unless we assume the axiom of choice (AC). Several relations of implication and independence are established. In the process, we answer a few open questions about the relations between different definitions of finiteness.
Forcing countable networks for spaces satisfying
We show that all finite powers of a Hausdorff space do not contain uncountable weakly separated subspaces iff there is a c.c.c poset such that in is a countable union of -dimensional subspaces of countable weight. We also show that this...
Forcing for hL and hd
The present paper addresses the problem of attainment of the supremums in various equivalent definitions of the hereditary density hd and hereditary Lindelöf degree hL of Boolean algebras. We partially answer two problems of J. Donald Monk [13, Problems 50, 54], showing consistency of different attainment behaviour and proving that (for the variants considered) this is the best result we can expect.
Forcing in a general setting
Forcing in the alternative set theory. I
The technique of forcing is developed for the alternative set theory (AST) and similar weak theories, where it can be used to prove some new independence results. There are also introduced some new extensions of AST.
Forcing in the alternative set theory. II
By the technique of forcing, some new independence results are proved for the alternative set theory (AST) and similar weak theories: The scheme of choice is independent both of AST and of second order arithmetic, axiom of constructibility is independent of AST plus schemes of choice.
Forcing tightness in products of fans
We prove two theorems that characterize tightness in certain products of fans in terms of families of integer-valued functions. We also define several notions of forcing that allow us to manipulate the structure of the set of functions from some cardinal θ to ω, and hence, the tightness of these products. These results give new constructions of first countable <θ-cwH spaces that are not ≤θ-cwH.
Forcing when there are large cardinals: An introduction
Forcing with proper classes
Forcing with propositional Lindenbaum algebras.
Forcings which preserve large cardinals