Abstract
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.
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References
Cohen, P.J., Set Theory and the Continuum Hypothesis, Benjamin, New York (1966).
Crawley, P., and Megibben, C., A simple construction of bizarre abelian groups, cited [F], 75. 1.
Devlin, K., The Yorkshireman’s guide to proper forcing, Proceeding of the 1978 Cambridge Set Theory Conference, to appear.
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.
Eklof, P., Set Theoretic Methods in Homoglocial Algebra and Abelian Groups les Presses de l’Université de Montréal (1980).
E2] Eklof, P., The structure of w1-separable groups, Trans. Amer. Math. Soc. (to appear).
Eklof, P. and Huber, M., On the rank of Ext. Math. Z. 174 (1980), 159–185.
EM] Eklof, P. and Mekler, A., On Endomorphism rings of w1-separable primary groups, this volume.
Fuchs, L., Infinite Abelian Groups volume II, Academic Press, New York (1973).
Hill,P., On the decomposition of groups, Can. J. Math. 21 (1969), 762–768.
Hu] Huber, M., Methods of set theory and the abundance of separable abelian p-groups, this volume.
Jech, T., Set Theory Academic Press, New York (1978).
Me] Megibben, C., Crawley’s problem on the unique w-elongation of p-groups is undecidable, Pacific J. Math. (to appear).
Mekler, A., How to construct almost free groups, Can. J. Math. 32 (1980) 1206–1228.
Mekler, A., Shelah’s Whitehead groups and CH, Rocky Mt. J. Math. 12 (1982) 271–278.
M3] Mekler, A., c.c.c. forcing without combinatorics, preprint.
M4] Mekler, A., Structure theory for w1-separable groups, in preparation.
Shelah, S., Infinite abelian groups, Whitehead problem and some constructions, Israel J. Math. 21 (1974) 243–256.
Shelah, S., On uncountable Abelian groups, Israel J. Math. 32 (1979), 311–330.
Shelah, S., Whitehead groups may not be free, even assuming CH, II, Israel J. Math. 35 (1980) 257–285.
Shelah, S., Proper Forcing Springer Verlag Lecture Notes in Math. 940 (1982).
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© 1983 Springer-Verlag Berlin Heidelberg
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Mekler, A.H. (1983). Proper Forcing and Abelian Groups. In: Göbel, R., Lady, L., Mader, A. (eds) Abelian Group Theory. Lecture Notes in Mathematics, vol 1006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21560-9_14
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DOI: https://doi.org/10.1007/978-3-662-21560-9_14
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