Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Journal of Applied Analysis

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

Managing Editor: Gajek, Leslaw

2 Issues per year


CiteScore 2016: 0.39

SCImago Journal Rank (SJR) 2015: 0.139
Source Normalized Impact per Paper (SNIP) 2015: 0.314

Mathematical Citation Quotient (MCQ) 2015: 0.13

Online
ISSN
1869-6082
See all formats and pricing
In This Section
Volume 10, Issue 2 (Jan 2004)

Properness Without Elementaricity

S. Shelah
  • The Hebrew University, Jerusalem, 91904, Israel and Mathematics Department, Rutgers University, New Brunswick, NJ 08854, USA. email:
Published Online: 2010-06-07 | DOI: https://doi.org/10.1515/JAA.2004.169

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

About the article

Received: 1997-12-23

Revised: 2003-05-05

Published Online: 2010-06-07

Published in Print: 2004-12-01



Citation Information: Journal of Applied Analysis, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908, DOI: https://doi.org/10.1515/JAA.2004.169. Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Saharon Shelah
Central European Journal of Mathematics, 2010, Volume 8, Number 2, Page 213
[2]
Jindřich Zapletal
Israel Journal of Mathematics, 2006, Volume 152, Number 1, Page 29
[3]
Vera Fischer, Sy David Friedman, and Yurii Khomskii
Archive for Mathematical Logic, 2014, Volume 53, Number 5-6, Page 695
[4]
Ilijas Farah and Saharon Shelah
Israel Journal of Mathematics, 2014, Volume 201, Number 2, Page 701
[5]
Heike Mildenberger and Saharon Shelah
Annals of Pure and Applied Logic, 2014, Volume 165, Number 2, Page 573
[6]
S. Shelah
Journal of Applied Analysis, 2006, Volume 12, Number 1, Page 1
[7]
Andrzej Rosłanowski and Saharon Shelah
Israel Journal of Mathematics, 2006, Volume 151, Number 1, Page 61

Comments (0)

Please log in or register to comment.
Log in