This paper investigates the combinatorial property of ultrafilters where the Mathias forcing relativized to them does not add dominating reals. We prove that the characterization due to Hrušák and Minami is equivalent to the strong P -point property. We also consistently construct a P -point that has no rapid Rudin-Keisler predecessor but that is not a strong P -point. These results answer questions of Canjar and Laflamme.

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. I wish to thank my advisor, Joel David Hamkins, for his constant support and guidance through my studies and the writing of this dissertation. It is safe to say that this text would not exist without his keen insight, openness to unusual ideas, and patience. It has been a pleasure and a privilege to work with him and I could not have asked for a better role model of mathematical ingenuity, rigour, and generosity. I would also like to thank Arthur Apter and Gunter Fuchs, the other members of my dissertation committee. Thank you both for your contributions, the many conversations, and for taking the time to read through this text. Thank you as well to Kameryn, Vika, Corey, the whole New York logic community, and the many other friends at the Graduate Center. I will remember fondly the hours spent talking about mathematics, playing games, and doing whatever other things graduate students do. Thank you, Kaethe. You have been (and hopefully remain) my best friend, sharing my happy moments and supporting me in the less happy ones. Thank you for letting me explain every part of this dissertation to you multiple times, and thank you for teaching me about mathematics that I would never have known or understood without you. I am incredibly fortunate to have met you and I can only hope to return your kindness and empathy in the future. Lastly, thank you to my parents and other family members who have stood by me and supported me through my long jo...

We investigate which filters on ω can contain towers, that is, a modulo finite descending sequence without any pseudointersection (in ${[\omega ]^\omega }$). We prove the following results:(1)Many classical examples of nice tall filters contain no towers (in ZFC).(2)It is consistent that tall analytic P-filters contain towers of arbitrary regular height (simultaneously for many regular cardinals as well).(3)It is consistent that all towers generate nonmeager filters (this answers a question of P. Borodulin-Nadzieja and D. Chodounský), in particular (consistently) Borel filters do not contain towers.(4)The statement “Every ultrafilter contains towers.” is independent of ZFC (this improves an older result of K. Kunen, J. van Mill, and C. F. Mills).Furthermore, we study many possible logical (non)implications between the existence of towers in filters, inequalities between cardinal invariants of filters (${\rm{ad}}{{\rm{d}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{co}}{{\rm{f}}^{\rm{*}}}\left( {\cal F} \right)$, ${\rm{no}}{{\rm{n}}^{\rm{*}}}\left( {\cal F} \right)$, and ${\rm{co}}{{\rm{v}}^{\rm{*}}}\left( {\cal F} \right)$), and the existence of Luzin type families (of size $\ge {\omega _2}$), that is, if ${\cal F}$ is a filter then ${\cal X} \subseteq {[\omega ]^\omega }$ is an ${\cal F}$-Luzin family if $\left\{ {X \in {\cal X}:|X \setminus F| = \omega } \right\}$ is countable for every $F \in {\cal F}$.

Forcing notions of the type P(ω)/I which do not add reals naturally add ultrafilters on ω. We investigate what classes of ultrafilters can be added in this way when I is a definable ideal. In particular, we show that if I is an F σ P-ideal the generic ultrafilter will be a P-point without rapid RK-predecessors which is not a strong P-point. This provides an answer to long standing open questions of Canjar and Laflamme.

In this short paper, we describe another class of forcing notions which preserve measurability of a large cardinal Ä from the optimal hypothesis, while adding new unbounded subsets to Ä. In some ways these forcings are closer to the Cohen-type forcings-we show that they are not minimal-but, they share some properties with treelike forcings. We show that they admit fusion-type arguments which allow for a uniform lifting argument.