Small Sets in Convex Geometry and Formal Independence over Zfc
نویسنده
چکیده
To each closed subset S of a finite dimensional Euclidean space corresponds a σ-ideal of sets J (S) which is σ-generated over S by the convex subsets of S. The set-theoretic properties of this ideal hold geometric information about the set. We discuss the relation of reducibility between convexity ideals and the connections between convexity ideals to other types of ideals, such as the ideals which are generated over squares of Polish space by graphs and inverses of graphs of continuous self maps, or Ramsey ideals, which are generated over Polish spaces by the homogeneous sets with respect to some continuous pair coloring. It is also attempted to present to non-specialists the set-theoretic methods for dealing with formal independence as a means for geometric investigations.
منابع مشابه
A convex combinatorial property of compact sets in the plane and its roots in lattice theory
K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...
متن کاملSweep Line Algorithm for Convex Hull Revisited
Convex hull of some given points is the intersection of all convex sets containing them. It is used as primary structure in many other problems in computational geometry and other areas like image processing, model identification, geographical data systems, and triangular computation of a set of points and so on. Computing the convex hull of a set of point is one of the most fundamental and imp...
متن کاملStatement Chris Lambie
My research lies mostly in logic and set theory, and in applications of set-theoretic tools to other areas of mathematics, such as graph theory, algebra, and topology. My set-theoretic work is largely combinatorial in nature and comes in one of two flavors: ZFC results and independence results. ZFC stands for Zermelo-Fraenkel axioms with choice and is the standard set of axioms in which set the...
متن کاملFUZZY HYPERVECTOR SPACES OVER VALUED FIELDS
In this note we first redefine the notion of a fuzzy hypervectorspace (see [1]) and then introduce some further concepts of fuzzy hypervectorspaces, such as fuzzy convex and balance fuzzy subsets in fuzzy hypervectorspaces over valued fields. Finally, we briefly discuss on the convex (balanced)hull of a given fuzzy set of a hypervector space.
متن کاملThe Hanf numbers of stationary logic II: Comparison with other logics
We show the ordering of the Hanf number of Lω,ω(wo), (well ordering) Lω,ω (quantification on countable sets), Lω,ω(aa) (stationary logic) and second order logic, has no more restraints provable in ZFC than previously known (those independence proofs assume CON(ZFC only). We also get results on corresponding logics for Lλ,μ. The author would like to thank the BSF and NSREC for partially supporti...
متن کامل