Dense ideals and cardinal arithmetic

From large cardinals we show the consistency of normal, fine, $\kappa$-complete $\lambda$-dense ideals on $\mathcal{P}_\kappa(\lambda)$ for successor $\kappa$. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman

Laver and set theory

In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip

Martin's maximum and the non-stationary ideal

We analyze the non-stationary ideal and the club filter at aleph_1 under MM

Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.Comment: Revised versio

Constructing regular ultrafilters from a model-theoretic point of view

This paper contributes to the set-theoretic side of understanding Keisler's order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality \lcf(\aleph_0, \de) of $\aleph_0$ modulo \de, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, detected by non-low theories. Assuming $\kappa > \aleph_0$ is measurable, we construct a regular ultrafilter on $\lambda \geq 2^\kappa$ which is flexible (thus: ok) but not good, and which moreover has large \lcf(\aleph_0) but does not even saturate models of the random graph. We prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. Finally, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on $SOP_2$. We prove that for any $n < \omega$, assuming the existence of $n$ measurable cardinals below $\lambda$, there is a regular ultrafilter $D$ on $\lambda$ such that any $D$-ultrapower of a model of linear order will have $n$ alternations of cuts, as defined below. Moreover, $D$ will $\lambda^+$-saturate all stable theories but will not $(2^{\kappa})^+$-saturate any unstable theory, where $\kappa$ is the smallest measurable cardinal used in the construction.Comment: 31 page

Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflectio