Large semilattices of breadth three

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice follows from either one of two axioms that are known to be independent of ZFC, namely (1) Martin's Axiom restricted to collections of aleph one dense subsets in posets of precaliber aleph one, (2) the existence of a gap-1 morass. In particular, the existence of such a lattice is consistent with ZFC, while the non-existence of such a lattice implies that omega two is inaccessible in the constructible universe. We also prove that for each regular uncountable cardinal $\kappa$ and each positive integer n, there exists a join-semilattice L with zero, of cardinality $\kappa^{+n}$ and breadth n+1, in which every principal ideal has less than $\kappa$ elements.Comment: Fund. Math., to appea

Bidimensional Inequalities with an Ordinal Variable

We investigate the normative foundations of two empirically implementable dominance criteria for comparing distributions of two attributes, where the first one is cardinal while the second is ordinal. The criteria we consider are Atkinson and Bourguignon\'s (1982) first quasi-ordering and a generalization of Bourguignon\'s (1989) ordered poverty gap criterion. In each case we specify the restrictions to be placed on the individual utility functions, which guarantee that all utility-inequality averse welfarist ethical observers will rank the distributions under comparison in the same way as the dominance criterion. We also identify the elementary inequality reducing transformations successive applications of which permit to derive the dominating distribution from the dominated one.Normative Analysis, Utilitarianism, Welfarism, Bidimensional Stochastic Dominance, Inequality Reducing Transformations

Capturing sets of ordinals by normal ultrapowers

We investigate the extent to which ultrapowers by normal measures on $\kappa$ can be correct about powersets $\mathcal{P}(\lambda)$ for $\lambda>\kappa$. We consider two versions of this questions, the capturing property $\mathrm{CP}(\kappa,\lambda)$ and the local capturing property $\mathrm{LCP}(\kappa,\lambda)$. $\mathrm{CP}(\kappa,\lambda)$ holds if there is an ultrapower by a normal measure on $\kappa$ which correctly computes $\mathcal{P}(\lambda)$. $\mathrm{LCP}(\kappa,\lambda)$ is a weakening of $\mathrm{CP}(\kappa,\lambda)$ which holds if every subset of $\lambda$ is contained in some ultrapower by a normal measure on $\kappa$. After examining the basic properties of these two notions, we identify the exact consistency strength of $\mathrm{LCP}(\kappa,\kappa^+)$. Building on results of Cummings, who determined the exact consistency strength of $\mathrm{CP}(\kappa,\kappa^+)$, and using a forcing due to Apter and Shelah, we show that $\mathrm{CP}(\kappa,\lambda)$ can hold at the least measurable cardinal.Comment: 20 page

Laver and set theory

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