A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals. (English) Zbl 0369.02034


03C60 Model-theoretic algebra
03E35 Consistency and independence results
03C75 Other infinitary logic
03E99 Set theory
20A10 Metamathematical considerations in group theory
05A05 Permutations, words, matrices
Full Text: DOI


[1] J. Baumgartner,A new kind of order types (preprint).
[2] C. C. Chang,Some Remarks on the Model Theory of Infinitary Languages; the Syntax and Semantics of Infinitary Languages, ed. Barwise, Springer-Verlag Lecture Notes No. 72, 1968, pp. 36–63.
[3] P. Eklof,Infinitary equivalence of abelian groups, Fund. Math., to appear. · Zbl 0327.02050
[4] P. Eklof,On the existence of {\(\kappa\)}-free abelian groups, to appear. · Zbl 0268.20036
[5] P. Eklof,Theorem of ZFCon abelian groups infinitarily equivalent to free groups, Notices Amer. Math. Soc.20 (1973), A-503.
[6] P. Erdös and A. Hajnal,Unsolved problems in set theory, Proc. Symp. Pure Math. XIII, Part I, Amer. Math. Soc., Providence, R. I., 1971, pp. 17–48. · Zbl 0228.04001
[7] P. Erdös and A. Hajnal,Unsolved and solved problems in set theory, Proc. Symp. in Honour of Tarski’s 70th Birthday, Berkeley, 1971; Proc. Symp. Pure Math. XXV, Amer. Math. Soc., Providence, R. I., 1974, 269–288.
[8] L. Fuchs,Infinite Abelian Groups, Vol. I, 1970, Vol. II, 1973, Academic Press, N. Y. and London. · Zbl 0209.05503
[9] W. Hanf,Incompactness in languages with infinitely long expressions, Fund. Math.53 (1963/64), 309–324. · Zbl 0207.30201
[10] P. Hill,New criteria for freeness in abelian groups, II. · Zbl 0296.20026
[11] P. Hill,On the freeness of abelian groups: a generalization of Pontryagin’s theorem, Bull. Amer. Math. Soc.76 (1970), 1118–1120. · Zbl 0223.20058
[12] P. Hill,A special criterion for freeness. · Zbl 0295.20061
[13] P. Hill,The splitting of modules and abelian groups, Canad. J. Math.
[14] G. Higman,Almost free groups, Proc. London Math. Soc.3, 1, (1951), 284–290. · Zbl 0043.25701
[15] J. Gregory,Abelian groups infinitarily equivalent to free ones, Notices Amer. Math. Soc.20 (1973), A-500.
[16] P. Griffith, n-free abelian groups, Aarhus University preprint series, 1971/72.
[17] R. L. Jensen,The fine structure of the constructible universe, Ann. Math. Logic4 (1972), 229–308. · Zbl 0257.02035
[18] A. Kurosch,Teoriya Grup, Moscow-Leningrad, 1944.
[19] A. Kurosch,The Theory of Groups, Chelsea Publ. Co., N. Y., 1960. · Zbl 0064.25104
[20] D. M. Kueker,Free and almost free algebra, Bull. Amer. Math. Soc., to appear. · Zbl 0204.31002
[21] E. Milner and S. Shelah,Two theorems on transversals, Proc. Symp. in Honour of Erdös’ 60th Birthday, Hungary, 1973, to appear. · Zbl 0322.05004
[22] E. Milner and S. Shelah,Sufficiency conditions for the existence of transversals, Canad. J. Math.26 (1974), 948–961. · Zbl 0303.05003
[23] N. Mirsky,Transversal Theory, Academic Press, New York, 1971.
[24] A. Mekler, Ph. D. thesis, Stanford University, in preparation.
[25] S. Shelah,Notes in partition calculus, Proc. Symp. in Honour of Erdös’ 60th Birthday, Hungary, 1973, to appear. · Zbl 0267.04006
[26] S. Shelah,Infinite abelian groups–Whitehead problem and some constructions, Israel J. Math.18 (1974), 243–256. · Zbl 0318.02053
[27] S. Shelah,Compactness in singular cardinalities, Notices Amer. Math. Soc.21 (1974), A-556.
[28] S. Shelah,Stability and Number of Non-isomorphic Models, North Holland Publ. Co., to appear. · Zbl 0713.03013
[29] S. Shelah,Incompactness in regular cardinals, in preparation. · Zbl 0617.03025
[30] S. Shelah,Various results in mathematical logic, Notices Amer. Math. Soc.22 (1975), A-23.
[31] S. Shelah,Various results in mathematical logic, Notices Amer. Math. Soc.22 (1975), A-474.
[32] E. Specker,Additive Gruppen von Fongen Ganzer Zahlen, Portugal. Math.9 (1950), 131–140. · Zbl 0041.36314
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.