MITCHELL-STYLE FORCING, WITH SMALL WORKING PARTS AND COLLECTIONS OF MODELS AS SIDE CONDITIONS, AND GAP-ONE SIMPLIFIED

Abstract. We give a modification of Mitchell’s technique ([8]-[11]) for adding objects of size ω2 with conditions with finite work-ing parts in which the collections of models used as side conditions are very highly structured, arguably making them more wieldy. We use one such forcing (essentially a ‘pure side conditions ’ forcing) to answer affirmatively the question, asked independently by Shelah and Velleman in the late 1980s, as to whether a (κ+, 1)-simplified morass can be added by a forcing with working parts of size < κ