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

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

Set Theory

This workshop included selected talks on pure set theory and its applications, simultaneously showing diversity and coherence of the subject

More Results on Regular Ultrafilters in ZFC

We prove, in ZFC alone, some new results on regularity and decomposability of ultrafilters. We also list some problems, and furnish applications to topological spaces and to extended logics.Comment: 57 page

Sums, numbers and infinity:Collections in Bolzano's mathematics and philosophy

This dissertation is a collection of essays almost exclusively focussing on Bernard Bolzano's theory of collections, and its connections to his conception of mathematics. The first significant contribution of this work is that it develops a novel approach for comparing and appraising Bolzano’s collections against the alternatives offered by set theory and mereology. This approach consists in focusing not on the metaphysical comparison but on the role Bolzano's collections play in his mathematics vis à vis the role the sets of set theory (ZFC) play for modern-day mathematics. The second important contribution of this dissertation is a novel interpretation of the interaction between Bolzano's theory of concepts on the one hand, and his way of comparing the size of infinite collections of natural numbers on the other. This culminates in a completely new interpretation of Bolzano's famous Paradoxes of the Infinite as a work that is not an anticipation of Cantorian set theory, but an attempt at deploying philosophical insights on the infinite to develop a sound treatment of infinite series, both converging and diverging ones