2017/10/30 by Peter Holy, Holy, Peter, Regula Krapf +3
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.1710.10826
openalex publication_date 2017/10/30 · openalex created_date 2022/10/02 · openalex updated_date 2026/08/01
We present three natural combinatorial properties for class forcing notions,\nwhich imply the forcing theorem to hold. We then show that all known sufficent\nconditions for the forcing theorem (except for the forcing theorem itself),\nincluding the three properties presented in this paper, imply yet another\nregularity property for class forcing notions, namely that proper classes of\nthe ground model cannot become sets in a generic extension, that is they do not\nhave set-sized names in the ground model. We then show that over certain models\nof G "odel-Bernays set theory without the power set axiom, there is a notion of\nclass forcing which turns a proper class into a set, however does not satisfy\nthe forcing theorem. Moreover, we show that the property of not turning proper\nclasses into sets can be used to characterize pretameness over such models of\nG "odel-Bernays set theory.\n