Cardinal arithmetic for skeptics

We present a survey of some results of the pcf-theory and their applications to cardinal arithmetic. We review basics notions (in section 1), briefly look at history in section 2 (and some personal history in section 3). We present main results on pcf in section 5 and describe applications to cardinal arithmetic in section 6. The limitations on independence proofs are discussed in section 7, and in section 8 we discuss the status of two axioms that arise in the new setting. Applications to other areas are found in section 9.Comment: 14 page

A Lower Bound for Generalized Dominating Numbers

We show a new proof for the fact that when $\kappa$ and $\lambda$ are infinite cardinals satisfying $\lambda ^ \kappa = \lambda$, the cofinality of the set of all functions from $\lambda$ to $\kappa$ ordered by everywhere domination is $2^\lambda$. An earlier proof was a consequence of a result about independent families of functions. The new proof follows directly from the main theorem we present: for every $A \subseteq \lambda$ there is a function $f: {^\kappa \lambda} \to \kappa$ such that whenever $M$ is a transitive model of $\textrm{ZF}$ such that ${^\kappa \lambda} \subseteq M$ and some $g: {^\kappa \lambda} \to \kappa$ in $M$ dominates $f$, then $A \in M$. That is, "constructibility can be reduced to domination".Comment: 9 page

On Singular Stationarity I (mutual stationarity and ideal-based methods)

We study several ideal-based constructions in the context of singular stationarity. By combining methods of strong ideals, supercompact embeddings, and Prikry-type posets, we obtain three consistency results concerning mutually stationary sets, and answer a question of Foreman and Magidor concerning stationary sequences on the first uncountable cardinals, $\aleph_n$, $1 \leq n < \omega$

Selected methods for the classification of cuts, and their applications

We consider four approaches to the analysis of cuts in ordered abelian groups and ordered fields, their interconnection, and various applications. The notions we discuss are: ball cuts, invariance group, invariance valuation ring, and cut cofinality

Menas' result is best possible

Generalizing some earlier techniques due to the second author, we show that Menas' theorem which states that the least cardinal kappa which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2^kappa supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpublished theorem due to the first author

Set theory without choice: not everything on cofinality is possible

We prove (ZF+DC) e.g. : if mu =|H(mu)| then mu^+ is regular non measurable. This is in contrast with the results for mu = aleph_{omega} on measurability see Apter Magidor [ApMg

The linear refinement number and selection theory

The \emph{linear refinement number} $\mathfrak{lr}$ is the minimal cardinality of a centered family in $[\omega]^\omega$ such that no linearly ordered set in $([\omega]^\omega,\subseteq^*)$ refines this family. The \emph{linear excluded middle number} $\mathfrak{lx}$ is a variation of $\mathfrak{lr}$. We show that these numbers estimate the critical cardinalities of a number of selective covering properties. We compare these numbers to the classic combinatorial cardinal characteristics of the continuum. We prove that $\mathfrak{lr}=\mathfrak{lx}=\mathfrak{fd}$ in all models where the continuum is at most $\aleph_2$, and that the cofinality of $\mathfrak{lr}$ is uncountable. Using the method of forcing, we show that $\mathfrak{lr}$ and $\mathfrak{lx}$ are not provably equal to $\mathfrak{d}$, and rule out several potential bounds on these numbers. Our results solve a number of open problems

Perfect Tree Forcings for Singular Cardinals

We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $\langle \kappa_n: n< \omega \rangle$, Prikry defined the forcing $\mathbb{P}$ all perfect subtrees of $\prod_{n<\omega}\kappa_n$, and proved that for $\kappa=\sup_{n<\omega}\kappa_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(\omega,\mu)$-distributive for all $\mu<\kappa$ but $(\omega,\kappa,\delta)$-distributivity fails for all $\delta<\kappa$, implying failure of the $(\omega,\kappa)$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. $\mathbb{P}$ satisfies a Sacks-type property, implying that $\mathbb{B}$ is $(\omega,\infty,<\kappa)$-distributive. The $(\mathfrak{h},2)$-d.l. and the $(\mathfrak{d},\infty,<\kappa)$-d.l. fail in $\mathbb{B}$. \mathcal{P}(\omega)/\mbox{Fin} completely embeds into $\mathbb{B}$. Also, $\mathbb{B}$ collapses $\kappa^\omega$ to $\mathfrak{h}$. We further prove that if $\kappa$ is a limit of countably many measurable cardinals, then $\mathbb{B}$ adds a minimal degree of constructibility for new $\omega$-sequences. Some of these results generalize to cardinals $\kappa$ with uncountable cofinality.Comment: 26 page

Viva la difference I: Nonisomorphism of ultrapowers of countable models

We show that it is not provable in ZFC that any two countable elementarily equivalent structures have isomorphic ultrapowers relative to some ultrafilter on omega

Exactly controlling the non-supercompact strongly compact cardinals

We summarize the known methods of producing a non-supercompact strongly compact cardinal and describe some new variants. Our Main Theorem shows how to apply these methods to many cardinals simultaneously and exactly control which cardinals are supercompact and which are only strongly compact in a forcing extension. Depending upon the method, the surviving non-supercompact strongly compact cardinals can be strong cardinals, have trivial Mitchell rank or even contain a club disjoint from the set of measurable cardinals. These results improve and unify previous results of the first author.Comment: 30 pages. To appear in the Journal of Symbolic Logi