<p>Square principles are statements about an important class of infinitary combinatorial objects. They may hold or fail to hold at singular cardinals depending on our large cardinal assumptions, but their precise consistency strengths are not yet known. </p><p> In this paper I present two theorems which greatly lower the known upper bounds of the consistency strengths of the failure of several square principles at singular cardinals. I do this using forcing constructions. First, using a quasicompact* cardinal I construct a model of the failure of ¬□([special characters omitted], < ω). Second, using a cardinal which is both subcompact and measurable, I construct a model of □<sub>κ,2</sub> + ¬□<sub> κ</sub> in which κ is singular. This paves the way for several natural extensions of these results.</p>
| Identifer | oai:union.ndltd.org:PROQUEST/oai:pqdtoai.proquest.com:3597431 |
| Date | 28 December 2013 |
| Creators | Holben, Ryan |
| Publisher | University of California, Irvine |
| Source Sets | ProQuest.com |
| Language | English |
| Detected Language | English |
| Type | thesis |
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