Nonelementary proper forcing notions
Abstract
We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(chi), in). This leads to forcing notions which are ``reasonably'' definable. We present two specific properties materializing this intuition: nep (nonelementary properness) and snep (Souslin nonelementary properness). For this we consider candidates (countable models to which the definition applies), and the older Souslin proper. A major theme here is ``preservation by iteration'', but we also show a dichotomy: if such forcing notions preserve the positiveness of the set of old reals for some naturally define c.c.c. ideals, then they preserve the positiveness of any old positive set. We also prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 December 1997
 arXiv:
 arXiv:math/9712283
 Bibcode:
 1997math.....12283S
 Keywords:

 Mathematics  Logic