Essential Kurepa trees versus essential Jech-Kunen trees

By an ! 1 #tree we mean a tree of cardinality ! 1 and height ! 1 . An ! 1 #tree is called a Kurepa tree if all its levels are countable and it has more than ! 1 branches. An ! 1 #tree is called a Jech#Kunen tree if it has # branches for some # strictly between ! 1 and 2 !1 . A Kurepa tree is called an essential Kurepa tree if it contains no Jech#Kunen subtrees. A Jech#Kunen tree is called an essential Jech#Kunen tree if it contains no Kurepa subtrees. In this paper we prove that #1# it is consistent with CH and 2 !1 # ! 2 that there exist essential Kurepa trees and there are no essential Jech#Kunen trees, #2# it is consistent with CH and 2 !1 # ! 2 plus the existence of a Kurepa tree with 2 !1 branches that there exist essential Jech#Kunen trees and there are no essential Kurepa trees. In the second result we require the existence of a Kurepa tree with 2 !1 branches in order to avoid triviality. 0 Introduction Our trees are always growing downward. We use T # for the # th ..

The Universality Problem

The theme of this thesis is to explore the universality problem in set theory in connection to model theory, to present some methods for finding universality results, to analyse how these methods were applied, to mention some results and to emphasise some philosophical interrogations that these aspects entail. A fundamental aspect of the universality problem is to find what determines the existence of universal objects. That means that we have to take into consideration and examine the methods that we use in proving their existence or nonexistence, the role of cardinal arithmetic, combinatorics etc. The proof methods used in the mathematical part will be mostly set-theoretic, but some methods from model theory and category theory will also be present. A graph might be the simplest, but it is also one of the most useful notions in mathematics. We show that there is a faithful functor F from the category L of linear orders to the category G of graphs that preserves model theoretic-related universality results (classes of objects having universal models in exactly the same cardinals, and also having the same universality spectrum). Trees constitute combinatorial objects and have a central role in set theory. The universality of trees is connected to the universality of linear orders, but it also seems to present more challenges, which we survey and present some results. We show that there is no embedding between an ℵ2-Souslin tree and a non-special wide ℵ2 tree T with no cofinal branches. Furthermore, using the notion of ascent path, we prove that the class of non-special ℵ2-Souslin tree with an ω-ascent path a has maximal complexity number, 2ℵ2 = ℵ3. Within the general framework of the universality problem in set theory and model theory, while emphasising their approaches and their connections with regard to this topic, we examine the possibility of drawing some philosophical conclusions connected to, among others, the notions of mathematical knowledge, mathematical object and proof

From axiomatization to generalizatrion of set theory

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results

Topics in Topology and Homotopy Theory

This book is an account of certain topics in general and algebraic topology