Related papers: Cardinal characteristics of the continuum and part…
We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orders which we call…
We introduce combinatorial principles that characterize strong compactness and supercompactness for inaccessible cardinals but also make sense for successor cardinals. Their consistency is established from what is supposedly optimal.…
We lay the combinatorial foundations for [ShSt:340] by setting up and proving the essential properties of the coding apparatus for singular cardinals. We also prove another result concerning the coding apparatus for inaccessible cardinals.
Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values.
Let P be the direct product of countably many copies of the additive group Z of integers. We study, from a set-theoretic point of view, those subgroups of P for which all homomorphisms to Z annihilate all but finitely many of the standard…
We discuss two general aspects of the theory of cardinal characteristics of the continuum, especially of proofs of inequalities between such characteristics. The first aspect is to express the essential content of these proofs in a way that…
We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (boldface Sigma^0_2 and, under suitable restrictions, Pi^0_2)…
Cardinal characteristics of the continuum represent the boundaries in size between the countable and the continuum with respect to certain properties of sets. They are often defined as the minimum sizes of families of reals that meet some…
We explore various combinatorial problems mostly borrowed from physics, that share the property of being continuously or discretely integrable, a feature that guarantees the existence of conservation laws that often make the problems…
Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (tau-covers). We deal with two types of combinatorial questions which arise from this study. 1.…
We consider several notions of well-foundedness of cardinals in the absence of the Axiom of Choice. Some of these have been conflated by some authors, but we separate them carefully. We then consider implications among these, and also…
For $\kappa$ regular and uncountable we define variants of the classical cardinal characteristics modulo the non-stationary ideal.
We have found some patterns in some triangles.
This article is an introduction to combinatorics under the axiom of determinacy with a focus on partition properties and infinity Borel codes.
We study the general problem of the behaviour of the continuum function in the presence of non-supercompact strongly compact cardinals.
We prove that combinatorial demand functions are characterized by two properties: continuity and the law of demand.
The manuscript contains elegant extensions of the fundamental variational principles: Br\o ndsted's and Ekeland's. On the other hand we get general and precise version of the Takahashi and the Cariste fixed point theorems. The results are…
We study selective and game-theoretic versions of properties like the ccc, weak Lindel\"ofness and separability, giving various characterizations of them and exploring connections between these properties and some classical cardinal…
Some physical consequences of the negation of the continuum hypothesis are considered. It is shown that quantum and classical mechanics are component parts of the multicomponent description of the set of variable infinite cardinality.…
We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.