Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The yellow cake

Authors: Andrzej Rosłanowski and Saharon Shelah
Journal: Proc. Amer. Math. Soc. 129 (2001), 279-291
MSC (2000): Primary 03E17, 03E35, 03E50
Published electronically: June 14, 2000
MathSciNet review: 1694876
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the following property: $(\circledast ^{\mathrm {Da}})$ For every function $f:\mathbb {R} \times \mathbb {R}\longrightarrow \mathbb {R}$ there are functions $g^0_n$, $g^1_n:\mathbb {R}\longrightarrow \mathbb {R}$ (for $n<\omega$) such that \[ (\forall x,y\in \mathbb {R})(f(x,y)=\sum _{n<\omega }g^0_n(x)g^1_n(y)).\] We show that, despite some expectation suggested by S. Shelah (1997), $(\circledast ^{\mathrm {Da}})$ does not imply $\mathbf {MA}(\sigma \mbox {-centered})$. Next, we introduce cardinal characteristics of the continuum responsible for the failure of $(\circledast ^{\mathrm {Da}})$.

References [Enhancements On Off] (What's this?)

  • Tomek Bartoszyński and Haim Judah, Set theory, A K Peters, Ltd., Wellesley, MA, 1995. On the structure of the real line. MR 1350295
  • Krzysztof Ciesielski. Set theoretic real analysis. Journal of Applied Analysis, 3:143-190, 1997.
  • Roy O. Davies, Representation of functions of two variables as sums of rectangular functions. I, Fund. Math. 85 (1974), no. 2, 177–183. MR 346108, DOI
  • Kenneth Kunen. Inaccessibility properties of cardinals. PhD thesis, Stanford University, 1968.
  • Arnold W. Miller, Arnie Miller’s problem list, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 645–654. MR 1234292
  • Arnold W. Miller. Some interesting problems. Circulated notes; available at
  • Saharon Shelah. Whitehead groups may not be free, even assuming CH. II. Israel Journal of Mathematics, 35:257-285, 1980 .
  • Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
  • Saharon Shelah. On Ciesielski’s Problems. Journal of Applied Analysis, 3(2):191-209, 1997.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 03E17, 03E35, 03E50

Retrieve articles in all journals with MSC (2000): 03E17, 03E35, 03E50

Additional Information

Andrzej Rosłanowski
Affiliation: Department of Mathematics and Computer Science, Boise State University, Boise, Idaho 83725 and Mathematical Institute of Wroclaw University, 50384 Wroclaw, Poland
Address at time of publication: Department of Mathematics, University of Nebraska at Omaha, Omaha, Nebraska 68182
MR Author ID: 288334

Saharon Shelah
Affiliation: Institute of Mathematics, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, and Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08854
MR Author ID: 160185
ORCID: 0000-0003-0462-3152

Received by editor(s): September 25, 1998
Received by editor(s) in revised form: March 31, 1999
Published electronically: June 14, 2000
Additional Notes: The first author thanks the Hebrew University of Jerusalem for support during his visit to Jerusalem in Summer ’98 when most of this research was done and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114.
The research of the second author was partially supported by The Israel Science Foundation. Publication 686.
Communicated by: Carl G. Jockusch
Article copyright: © Copyright 2000 American Mathematical Society