Flipping properties and huge cardinals
Flipping properties and supercompact cardinals
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 indestructible extensions of almost disjoint families
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 ideals generated by closed sets
Consider the poset where is an arbitrary -ideal -generated by a projective collection of closed sets. Then the extension is given by a single real of an almost minimal degree: every real is Cohen-generic over or .
Forcing with proper classes
Forcing with propositional Lindenbaum algebras.
Forcings which preserve large cardinals
Fragments of strong compactness, families of partitions and ideal extensions
We investigate some natural combinatorial principles related to the notion of mild ineffability, and use them to obtain new characterizations of mild ineffable and weakly compact cardinals. We also show that one of these principles may be satisfied by a successor cardinal. Finally, we establish a version for of the canonical Ramsey theorem for pairs.
Fraïssé structures and a conjecture of Furstenberg
We study problems concerning the Samuel compactification of the automorphism group of a countable first-order structure. A key motivating question is a problem of Furstenberg and a counter-conjecture by Pestov regarding the difference between , the Samuel compactification, and , the enveloping semigroup of the universal minimal flow. We resolve Furstenberg’s problem for several automorphism groups and give a detailed study in the case of , leading us to define and investigate several new types...
Free -products of lattices. I
Free -products of lattices. II
Free Power or Width of some Kinds of Mathematical Structures
From an alternative model of rough sets to fuzzy sets