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ISSN 1088-6826(e) ISSN 0002-9939(p)

Ideals without ccc and without property $\boldsymbol( \mathbf{M} \boldsymbol)$

Author(s): Howard Becker
Journal: Proc. Amer. Math. Soc. 128 (2000), 3031-3034.
MSC (2000): Primary 03E15
Posted: April 28, 2000
MathSciNet review: 1670422
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Abstract | References | Similar articles | Additional information

Abstract: We prove a strong version of a theorem of Balcerzak-Roslanowski-Shelah by showing, in ZFC, that there exists a simply definable Borel $\sigma$ -ideal for which both the ccc and property (M) fail. The proof involves Polish group actions.

References:

1.: M. Balcerzak, Can ideals without ccc be interesting?, Topology and its Applications 55 (1994), 251-260. MR 95g:54032
2.: M. Balcerzak, A. Roslanowski and S. Shelah, Ideals without ccc, Journal of Symbolic Logic 63 (1998), 128-148. MR 99j:03039
3.: H. Becker, The topological Vaught's conjecture and minimal counterexamples, Journal of Symbolic Logic 59 (1994), 757-784. MR 95k:03077
4.: H. Becker and A.S. Kechris, The Descriptive Set Theory of Polish Group Actions, Cambridge University Press, 1996. MR 98d:54068
5.: H. Friedman, Countable models of set theories, Cambridge Summer School in Mathematical Logic (A.R.D. Mathias and H. Rogers, eds.) (1973), 539-573. MR 50:102
6.: A.S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995. MR 96e:03057
7.: H.J. Keisler, Model Theory for Infinitary Logic, North-Holland, 1971. MR 49:8855
8.: Y.N. Moschovakis, Descriptive Set Theory, North-Holland, 1980. MR 82e:03002

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Additional Information:

Howard Becker
Affiliation: Department of Mathematics, The University of South Carolina, Columbia, South Carolina 29208
Email: becker@math.sc.edu

DOI: 10.1090/S0002-9939-00-05373-9
PII: S 0002-9939(00)05373-9
Keywords: Ideal of sets, Polish group action
Received by editor(s): October 23, 1998
Received by editor(s) in revised form: December 3, 1998
Posted: April 28, 2000
Additional Notes: The author's research was partially supported by NSF Grant DMS-9505505.
Communicated by: Carl G. Jockusch, Jr.
Copyright of article: Copyright 2000, American Mathematical Society

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