Physical Sciences and Mathematics Commons^™

Tail-Measurable Functions And Their Corresponding Induced Classes, And Some Determinacy Conditions Involving 3-Player Games, Joshua K. Reagan

UNLV Theses, Dissertations, Professional Papers, and Capstones

In this dissertation, we have two main categories of results. The first is regarding certain point-classes, and the second is regarding 3-player games.

The point-classes of Baire Space, \mathcal{N}, in the Borel and Projective Hierarchies, as well as Hausdorff's Difference Hierarchy have been well studied, and there has been much research into further stratifying these hierarchies. One area of particular interest falls in between the point-classes \mathbf{\Pi}_\mathbf{1}^\mathbf{1} and \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). It is well known that the point-classes \beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}, for \beta\in\omega^2, stratify this region of the projective hierarchy, with the point-class \bigcup_{\beta\in\omega^2}\beta-\mathbf{\Pi}_\mathbf{1}^\mathbf{1} still falling strictly below \Delta\left(\omega^2-\mathbf{\Pi}_\mathbf{1}^\mathbf{1}\right). Dr. Derrick DuBose developed multiple …

Exploring The Choiceless Cardinal Hierarchy, David Linkletter

UNLV Theses, Dissertations, Professional Papers, and Capstones

In 1971, Kunen proved that the Axiom of Choice imposes a ceiling on the large cardinal hierarchy [7]. Much like the assumption V ≠ L unlocks measurable cardinals and beyond, dropping the Axiom of Choice enables Reinhardt cardinals and stronger cardinals to be explored. Some major notions of large cardinals beyond choice have recently been standardized by Woodin et. al. [2], with questions raised regarding their interconnectedness. Part 1 of this dissertation partially answers two of those questions, while conjecturing, with a partial solution, a much stronger answer which would simplify the existing cardinal charts - that Regular Berkeley Cardinals …