### Properness Without Elementaricity

by Shelah. [Sh:630]

J Applied Analysis, 2004

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.

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