The purpose of this thesis is to add to the structure theory of Aronszajn orderings. We shall focus essentially in four topics. The first topic of discussion is about the relation between Lipschitz and coherent trees. I will demonstrate that the tree $T(\rho_0)$ is coherent without any extra set theoretic hypothesis. The second topic presents an application of Todorcevic's $\rho$ functions to provide some partial answers to an old question of Juhaz asking whether a standard weakening of Jensen's diamond principle implies the existence of Suslin trees. In the third topic we focus on providing a satisfactory rough classification result of the class of Aronszajn lines. Our main result is that, assuming PFA, the class of Aronszajn lines is well-quasi-ordered by embeddability. The last topic is an investigation of the gap structure of the class of coherent Aronszajn trees. I will show that, assuming PFA, the class of coherent Aronszajn trees quasi-ordered by embeddability is the unique saturated linear order of cardinality $\aleph_2$.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/29807 |
| Date | 31 August 2011 |
| Creators | Martinez-Ranero, Carlos Azarel |
| Contributors | Todorcevic, Stevo |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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