MITCHELL-INSPIRED FORCING, WITH SMALL WORKING PARTS AND COLLECTIONS OF MODELS OF UNIFORM SIZE AS SIDE CONDITIONS, AND GAP-ONE SIMPLIFIED MORASSES

Abstract

AbstractWe show that a $(\kappa ^{+},1)$ -simplified morass can be added by a forcing with working parts of size smaller than $\kappa $ . This answers affirmatively the question, asked independently by Shelah and Velleman in the early 1990s, of whether it is possible to do so.Our argument use a modification of a technique of Mitchell’s for adding objects of size $\omega _2$ in which collections of models – all of equal, countable size – are used as side conditions. In our modification, whilst the individual models are, as in Mitchell’s technique, taken ad hoc from quite general classes, the collections of models are very highly structured, in a way that is somewhat different from, perhaps more stringent than, Mitchell’s original, arguably making the method more wieldy and giving the prospect of further uses with more delicate working parts.