Proper Forcing and Abelian Groups

  • Alan H. Mekler
Part of the Lecture Notes in Mathematics book series (LNM, volume 1006)


It is well known that assuming Martin’s axiom and the negation of the continuum hypothesis (MA+−1CH) has a dramatic effect on Abelian group theory. One only has to think of Shelah’s resolution of the Whitehead problem (cf[S1] or [E1]). MA+−NCH has certain drawbacks. It is not strong enough to resolve questions about the structure of ω1-separable groups which can be resolved by an extension and it is not consistent with the continuum hypothesis. In this paper I shall talk about generalizations due to Shelah of MA+−ICH, the proper forcing axioms, and their relation to Abelian group theory. (From hereon “group” will mean “Abelian group”.) Section I is devoted to some preliminaries and the statement of the proper forcing axioms. In section II a useful result about choosing a cub almost disjoint from a ladder system assuming PFA(ω1) is proved in order to illustrate how the axiom works. Proper posets are redefined in section III using elementary submodels. This section simplifies the task of writing the remainder of the paper. But it is explained for the reader who is unversed in logic how to ignore this section.


Direct Summand Dense Subset Continuum Hypothesis Pure Subgroup Maximal Antichain 
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  1. [C]
    Cohen, P.J., Set Theory and the Continuum Hypothesis, Benjamin, New York (1966).Google Scholar
  2. [CM]
    Crawley, P., and Megibben, C., A simple construction of bizarre abelian groups, cited [F], 75. 1.Google Scholar
  3. [D]
    Devlin, K., The Yorkshireman’s guide to proper forcing, Proceeding of the 1978 Cambridge Set Theory Conference, to appear.Google Scholar
  4. [DS]
    Devlin, K. and Shelah, S., A weak version of Q which follows from K 2 0 2 1, Israel J. Math. 29 (1978), 239–247.Google Scholar
  5. [E1]
    Eklof, P., Set Theoretic Methods in Homoglocial Algebra and Abelian Groups les Presses de l’Université de Montréal (1980).Google Scholar
  6. E2] Eklof, P., The structure of w1-separable groups, Trans. Amer. Math. Soc. (to appear).Google Scholar
  7. [EH]
    Eklof, P. and Huber, M., On the rank of Ext. Math. Z. 174 (1980), 159–185.CrossRefGoogle Scholar
  8. EM] Eklof, P. and Mekler, A., On Endomorphism rings of w1-separable primary groups, this volume.Google Scholar
  9. [F]
    Fuchs, L., Infinite Abelian Groups volume II, Academic Press, New York (1973).Google Scholar
  10. [H]
    Hill,P., On the decomposition of groups, Can. J. Math. 21 (1969), 762–768.Google Scholar
  11. Hu] Huber, M., Methods of set theory and the abundance of separable abelian p-groups, this volume.Google Scholar
  12. [J]
    Jech, T., Set Theory Academic Press, New York (1978).Google Scholar
  13. Me] Megibben, C., Crawley’s problem on the unique w-elongation of p-groups is undecidable, Pacific J. Math. (to appear).Google Scholar
  14. [Ml]
    Mekler, A., How to construct almost free groups, Can. J. Math. 32 (1980) 1206–1228.Google Scholar
  15. [M2]
    Mekler, A., Shelah’s Whitehead groups and CH, Rocky Mt. J. Math. 12 (1982) 271–278.Google Scholar
  16. M3] Mekler, A., c.c.c. forcing without combinatorics, preprint.Google Scholar
  17. M4] Mekler, A., Structure theory for w1-separable groups, in preparation.Google Scholar
  18. [S1]
    Shelah, S., Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 21 (1974) 243–256.Google Scholar
  19. [S2]
    Shelah, S., On uncountable Abelian groups, Israel J. Math. 32 (1979), 311–330.Google Scholar
  20. [S3]
    Shelah, S., Whitehead groups may not be free, even assuming CH, II, Israel J. Math. 35 (1980) 257–285.Google Scholar
  21. [S4]
    Shelah, S., Proper Forcing Springer Verlag Lecture Notes in Math. 940 (1982).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Alan H. Mekler
    • 1
  1. 1.Research supported by Natural ScienceEngineering Council of CanadaCanada

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