161 research outputs found
The embedding structure for linearly ordered topological spaces
In this paper, the class of all linearly ordered topological spaces (LOTS)
quasi-ordered by the embeddability relation is investigated. In ZFC it is
proved that for countable LOTS this quasi-order has both a maximal (universal)
element and a finite basis. For the class of uncountable LOTS of cardinality
it is proved that this quasi-order has no maximal element for
at least the size of the continuum and that in fact the dominating number for
such quasi-orders is maximal, i.e. . Certain subclasses of LOTS, such
as the separable LOTS, are studied with respect to the top and internal
structure of their respective embedding quasi-order. The basis problem for
uncountable LOTS is also considered; assuming the Proper Forcing Axiom there is
an eleven element basis for the class of uncountable LOTS and a six element
basis for the class of dense uncountable LOTS in which all points have
countable cofinality and coinitiality
The Infinite as Method in Set Theory and Mathematics
Este artÃculo da cuenta de la aparición histórica de lo infinito en la teorÃa de conjuntos, y de cómo lo tratamos dentro y fuera de las matemáticas. La primera sección analiza el surgimiento de lo infinito como una cuestión de método en la teorÃa de conjuntos. La segunda sección analiza el infinito dentro y fuera de las matemáticas, y cómo deben adoptarse.This article address the historical emergence of the infinite in set theory, and how we are to take the infinite in and out of mathematics.The first section discusses the emergence of the infinite as a matter of method in set theory. The second section discusses the infinite in and out of mathematics, and how it is to be taken
Hilbert Spaces Without Countable AC
This article examines Hilbert spaces constructed from sets whose existence is
incompatible with the Countable Axiom of Choice (CC). Our point of view is
twofold: (1) We examine what can and cannot be said about Hilbert spaces and
operators on them in ZF set theory without any assumptions of Choice axioms,
even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some
set-theoretic results from associated Hilbert spaces.Comment: 51 page
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
Classical Set Theory: Theory of Sets and Classes
This is a short introductory course to Set Theory, based on axioms of von
Neumann--Bernays--G\"odel (briefly NBG). The text can be used as a base for a
lecture course in Foundations of Mathematics, and contains a reasonable minimum
which a good (post-graduate) student in Mathematics should know about
foundations of this science.Comment: 162 page
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