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Journal of Applied Analysis

Editor-in-Chief: Liczberski, Piotr / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

2 Issues per year

Mathematical Citation Quotient 2013: 0.13

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Properness Without Elementaricity

S. Shelah1

1The Hebrew University, Jerusalem, 91904, Israel and Mathematics Department, Rutgers University, New Brunswick, NJ 08854, USA. email:

Citation Information: Journal of Applied Analysis. Volume 10, Issue 2, Pages 169–289, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908, DOI: 10.1515/JAA.2004.169, June 2010

Publication History

Received:
1997-12-23
Revised:
2003-05-05
Published Online:
2010-06-07

Abstract

We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary sub-models of some (H(χ), ∈). 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) and also the older Souslin proper. For this we consider candidates (countable models to which the definition applies). 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 defined c.c.c. ideal, then they preserve the positiveness of any old positive set hence preservation by composition of two follows. Last but not least, we prove that (among such forcing notions) the only one commuting with Cohen is Cohen itself; in other words, any other such forcing notion make the set of old reals to a meager set. In the end we present some open problems in this area.

Key words and phrases.: set theory; forcing; iterated forcing; preservation for iterated forcing; nep; snep; Cohen forcing; commuting forcing; Fubini theorem

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