Name principles, and hierarchies of regular cardinals applied to LST numbers and inner model theory

Abstract

We present several new discoveries in two multiple areas of set theory. First, we introduce a new kind of forcing axiom known as a “name principle”. We give a detailed and comprehensive account of how these name principles relate to one another and to the classical forcing axioms. This leads us to consider new variants of the traditional forcing axioms, and we also include these in our account. We then show several examples of uses for these relationships, including improvements to, and substantially simplified proofs of, several well-known theorems.

We then turn to another topic, looking at stratifying the class Reg of V regular cardinals by Cantor-Bendixson rank. Letting Reg_<α be the class of all elements of Reg of rank <α, we show (under fairly weak large cardinal assumptions) that L[Reg_<α] can be expressed as a generic extension of an iteration of a type of model called a “mouse”. We also prove a similar result for Reg^s, the class of strong inaccessibles. Then we go on to show that all the mice we’ve used exist in L[Reg^s].

Finally, we generalise a result of Magidor and Väänänen. We define two different predicates of second order logic which are related to Reg_<α in much the same way that the famous predicate I is related to Card. We then find a lower bound for the Löwenheim-Skolem-Tarski number of these two predicates (together with I), and show that this lower bound is optimal.

Date of Award	24 Jan 2023
Original language	English
Awarding Institution	University of Bristol
Supervisor	Philip D Welch (Supervisor)