Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

On what I do not understand (and have something to say): Part I

This is a non-standard paper, containing some problems in set theory I have in various degrees been interested in. Sometimes with a discussion on what I have to say; sometimes, of what makes them interesting to me, sometimes the problems are presented with a discussion of how I have tried to solve them, and sometimes with failed tries, anecdote and opinion. So the discussion is quite personal, in other words, egocentric and somewhat accidental. As we discuss many problems, history and side references are erratic, usually kept at a minimum (``see ... '' means: see the references there and possibly the paper itself). The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear

Club guessing and the universal models

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element

Laver and set theory

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

A family of trees with no uncountable branches

We construct a family of 2 ℵ1 trees of size ℵ1 and no uncountable branches that in a certain way codes all ω1sequences of infinite subsets of ω. This coding allows us to conclude that in the presence of the club guessing between ℵ1 and ℵ0, these trees are pairwise very different. In such circumstances we can also conclude that the universality number of the ordered class of trees of size ℵ1 with no uncountable branches under “metric-preserving ” reductions must be at least the continuum. From the topological point of view, the above results show that under the same assumptions there are 2 ℵ1 pairwise non-isometrically embeddable first countable ω1metric spaces with a strong non-ccc property, and that their universality number under isometric embeddings is at least the continuum. Without the non-ccc requirement, a family of 2 ℵ1 pairwise non-isometrically embeddable first countable ω1-metric spaces exists in ZFC by an earlier result of S. Todorčević. The set-theoretic assumptions mentioned above are satisfied in many natural models of set theory (such as the ones obtained after forcing by a ccc forcing over a model of ♦). We use a similar method to discuss trees of size κ with no uncountable branches, for any regular uncountable κ