Dissertations, Theses, and Capstone Projects

Date of Degree

6-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Joel David Hamkins

Committee Members

Arthur W. Apter

Gunter Fuchs

Subject Categories

Logic and Foundations | Set Theory

Keywords

Large cardinals, Laver functions, guessing principles, number of normal measures, forcing axioms

Abstract

In chapter 1 a notion of independence for diamonds and Laver diamonds is investigated. A sequence of Laver diamonds for κ is joint if for any sequence of targets there is a single elementary embedding j with critical point κ such that each Laver diamond guesses its respective target via j. In the case of measurable cardinals (with similar results holding for (partially) supercompact cardinals) I show that a single Laver diamond for κ yields a joint sequence of length κ, and I give strict separation results for all larger lengths of joint sequences. Even though the principles get strictly stronger in terms of direct implication, I show that they are all equiconsistent. This is contrasted with the case of θ-strong cardinals where, for certain θ, the existence of even the shortest joint Laver sequences carries nontrivial consistency strength. I also formulate a notion of jointness for ordinary ◊κ-sequences on any regular cardinal κ. The main result concerning these shows that there is no separation according to length and a single ◊κ-sequence yields joint families of all possible lengths. In chapter 2 the notion of a grounded forcing axiom is introduced and explored in the case of Martin's axiom. This grounded Martin's axiom, a weakening of the usual axiom, states that the universe is a ccc forcing extension of some inner model and the restriction of Martin's axiom to the posets coming from that ground model holds. I place the new axiom in the hierarchy of fragments of Martin's axiom and examine its effects on the cardinal characteristics of the continuum. I also show that the grounded version is quite a bit more robust under mild forcing than Martin's axiom itself.

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