In this dissertation we prove some results about the existence of families of Aronszajn trees on successors of regular cardinals which are pairwise not club isomorphic. The history of this topic begins with a theorem of Gaifman and Specker in the 1960s which asserts the existence from ZFC of many pairwise not isomorphic Aronszajn trees. Since that result was proven, the focus has turned to comparing Aronszajn trees with respect to isomorphisms on a club of levels, instead of on the entire tree. In the 1980s Abraham and Shelah proved that the Proper Forcing Axiom implies that any two Aronszajn trees on the first uncountable cardinal are club isomorphic. This theorem was generalized to higher cardinals in recent work of Krueger. Abraham and Shelah also proved that the opposite holds under diamond principles. In this dissertation we address the existence of pairwise not club isomorphic Aronszajn trees on higher cardinals from a variety of cardinal arithmetic and diamond principle assumptions. For example, on the successor of a regular cardinal, assuming GCH and the diamond principle on the critical cofinality, there exists a large collection of special Aronszajn trees such that any two of them do not contain club isomorphic subtrees.
| Identifer | oai:union.ndltd.org:unt.edu/info:ark/67531/metadc1707408 |
| Date | 08 1900 |
| Creators | Chavez, Jose |
| Contributors | Krueger, John, Jackson, Steve, Trang, Nam |
| Publisher | University of North Texas |
| Source Sets | University of North Texas |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | v, 94 pages, Text |
| Rights | Public, Chavez, Jose, Copyright, Copyright is held by the author, unless otherwise noted. All rights Reserved. |
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