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Maximal Saturated Linear Orders

Maximal Saturated Linear Orders

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Title: Maximal Saturated Linear Orders
Author: Kibedi, Francisco Guillermo Justo
Abstract: The goal of this dissertation is to prove two theorems related to a question posed by Felix Hausdorff in 1907 regarding pantachies, which are maximal linearly ordered subsets of the space of real-valued sequences partially ordered by eventual domination. Hausdorff's question was as follows: is there a pantachie containing no gaps of order type the first uncountable cardinal?

In Chapter 1, some terminology is defined, and Hausdorff's question about pantachies is explored. Some related work by other mathematicians is examined, both preceding and following Hausdorff's paper. In Chapter 2, relevant definitions and results about forcing, gaps, and saturated linear orders are collected. Chapter 3 contains the complete proof of the first theorem, namely, the consistency of the existence of a saturated Hausdorff pantachie in a model where the continuum hypothesis (CH) fails. Finally, in Chapter 4, a different method is used to prove a stronger result, namely, the consistency of the existence of a saturated Hausdorff pantachie in a model of Martin's Axiom along with the negation of CH. The appendix mentions a few related open questions and some partial answers.
Subject: Theoretical mathematics
Keywords: Set Theory
Cardinal
Cardinal Arithmetic
Independence
Generic
Genericity
Delta-system
Hausdorff
Felix Hausdorff
Pantachie
Maximal linear order
Maximal
Linear
Linear order
Order
Partial order
Saturated
Saturated linear order
Maximal saturated linear order
Saturation
Eventual domination
Forcing
Forcing Extension
Iterated Forcing
Finite-support iterated forcing
Gaps
Gap-filling
Pregaps
Cuts
Cut-filling
continuum
Continuum Hypothesis
Martin’s Axiom
CH
MA
Kunen
Partial Order
Countable chain condition
Ccc
Special gaps
Strong gaps
Suslin Trees
Aronszajn Trees
Paul du Bois-Reymond
Rates of convergence
Infinitesimal
Hyper-reals
Ultrapower
Ultrafilter
Ideal
Divergence
Divergence Ordering
Convergence
Almost Containment
Richard Laver
Investigations into Order Types
Universal
Universal linear order
Baumgartner
James Baumgartner
Kenneth Kunen
Woodin
Hugh Woodin
Proper Forcing Axiom
Uniform Dense
Special Trees
Specializer
Tree-Specializer
Tree-Specializing forcing
ccc-fillable
ccc-indestructible
Pre-caliber
Pre-caliber aleph 1
Model
Ground model
Countable transitive model
ZFC
Antichain
Maximal antichain
Nice partial order
Kibedi
Francisco Kibedi
Steprans
Juris Steprans
Type: Electronic Thesis or Dissertation
Rights: Author owns copyright, except where explicitly noted. Please contact the author directly with licensing requests.
URI: http://hdl.handle.net/10315/32107
Supervisor: Steprans, Juris
Degree: PhD - Doctor of Philosophy
Program: Mathematics & Statistics
Exam date: 2015-09-24
Publish on: 2016-09-20

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