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Uncountable constructions for B.A. e.c. groups and banach spaces

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Abstract

This paper has two aims: to aid a non-logician to construct uncountable examples by reducing the problems to finitary problems, and also to present some construction solving open problems. We assume the diamond forℵ 1 and solve problems in Boolean algebras, existentially closed groups and Banach spaces. In particular, we show that for a given countable e.c. groupM there is no uncountable group embeddable in everyG \(G L_{L,\omega } \)-equivalent toM; and that there is a non-separable Banach space with noℵ 1 elements, no one being the closure of the convex hull of the others. Both had been well-known questions. We also deal generally with inevitable models (§4).

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Authors and Affiliations

  1. Department of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Sharon Shelah

  2. Department of Mathematics, University of Wisconsin, Madison, Wisconsin, USA

    Sharon Shelah

  3. Department of Mathematics, University of California, Berkeley, California, USA

    Sharon Shelah

Authors
  1. Sharon Shelah
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Additional information

The author would like to thank the NSF for partially supporting this research by grants H144-H747 and MCS-76-08479, and the United States-Israel Binational Science Foundation for partially supporting this research.

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Shelah, S. Uncountable constructions for B.A. e.c. groups and banach spaces. Israel J. Math. 51, 273–297 (1985). https://doi.org/10.1007/BF02764721

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  • DOI: https://doi.org/10.1007/BF02764721

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