Non-elementary 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 (non-elementary properness) and snep (Souslin non-elementary 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.