## When does almost free imply free? (For groups, transversals, etc.)

HTML articles powered by AMS MathViewer

- by Menachem Magidor and Saharon Shelah
- J. Amer. Math. Soc.
**7**(1994), 769-830 - DOI: https://doi.org/10.1090/S0894-0347-1994-1249391-8
- PDF | Request permission

## Abstract:

We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal $\kappa$ to ${\aleph _{\kappa + 1}}$. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for “ $\kappa$ free implies free”, and then we show, assuming the consistency of infinitely many supercompacts, that one can have a model of ZFC+G.C.H. in which ${\aleph _{{\omega ^2} + 1}}$ free implies ${\aleph _{{\omega ^2} + 2}}$ free. Similar construction yields a model in which ${\aleph _\kappa }$ free implies free for $\kappa$ the first cardinal fixed point (namely, the first cardinal $\alpha$ satisfying $\alpha = {\aleph _\alpha }$). The absolute results about the existence of almost free nonfree groups require only minimal knowledge of set theory. Also, no knowledge of metamathematics is required for reading the section on the combinatorial principle used to show that almost free implies free. The consistency of the combinatorial principle requires acquaintance with forcing techniques.## References

- James E. Baumgartner,
*A new class of order types*, Ann. Math. Logic**9**(1976), no. 3, 187–222. MR**416925**, DOI 10.1016/0003-4843(76)90001-2 - A. R. D. Mathias (ed.),
*Surveys in set theory*, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983. MR**823774**, DOI 10.1017/CBO9780511758867 - Shai Ben-David,
*On Shelah’s compactness of cardinals*, Israel J. Math.**31**(1978), no. 1, 34–56. MR**506381**, DOI 10.1007/BF02761379 - Maxim R. Burke and Menachem Magidor,
*Shelah’s $\textrm {pcf}$ theory and its applications*, Ann. Pure Appl. Logic**50**(1990), no. 3, 207–254. MR**1086455**, DOI 10.1016/0168-0072(90)90057-9
J. Cummings and H. Woodin, - Paul C. Eklof,
*On the existence of $\kappa$-free abelian groups*, Proc. Amer. Math. Soc.**47**(1975), 65–72. MR**379694**, DOI 10.1090/S0002-9939-1975-0379694-0 - Paul C. Eklof and Alan H. Mekler,
*Almost free modules*, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR**1055083** - P. Erdös and R. Rado,
*A partition calculus in set theory*, Bull. Amer. Math. Soc.**62**(1956), 427–489. MR**81864**, DOI 10.1090/S0002-9904-1956-10036-0 - Matthew Foreman and W. Hugh Woodin,
*The generalized continuum hypothesis can fail everywhere*, Ann. of Math. (2)**133**(1991), no. 1, 1–35. MR**1087344**, DOI 10.2307/2944324 - László Fuchs,
*Infinite abelian groups. Vol. I*, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR**0255673** - Moti Gitik,
*The negation of the singular cardinal hypothesis from $o(\kappa )=\kappa ^{++}$*, Ann. Pure Appl. Logic**43**(1989), no. 3, 209–234. MR**1007865**, DOI 10.1016/0168-0072(89)90069-9 - Phillip A. Griffith,
*Infinite abelian group theory*, University of Chicago Press, Chicago, Ill.-London, 1970. MR**0289638** - Graham Higman,
*Almost free groups*, Proc. London Math. Soc. (3)**1**(1951), 284–290. MR**44519**, DOI 10.1112/plms/s3-1.1.284 - Paul Hill,
*On the splitting of modules and abelian groups*, Canadian J. Math.**26**(1974), 68–77. MR**338217**, DOI 10.4153/CJM-1974-007-6 - Paul Hill,
*On the freeness of abelian groups: A generalization of Pontryagin’s theorem*, Bull. Amer. Math. Soc.**76**(1970), 1118–1120. MR**263919**, DOI 10.1090/S0002-9904-1970-12586-1 - Paul Hill,
*A special criterion for freeness*, Symposia Mathematica, Vol. XIII (Convegno di Gruppi Abeliani & Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972) Academic Press, London, 1974, pp. 311–314. MR**0360874** - Wilfrid Hodges,
*In singular cardinality, locally free algebras are free*, Algebra Universalis**12**(1981), no. 2, 205–220. MR**608664**, DOI 10.1007/BF02483879 - Thomas Jech,
*Set theory*, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR**506523** - R. Björn Jensen,
*The fine structure of the constructible hierarchy*, Ann. Math. Logic**4**(1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR**309729**, DOI 10.1016/0003-4843(72)90001-0 - A. Kanamori and M. Magidor,
*The evolution of large cardinal axioms in set theory*, Higher set theory (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1977) Lecture Notes in Math., vol. 669, Springer, Berlin, 1978, pp. 99–275. MR**520190** - Richard Laver,
*Making the supercompactness of $\kappa$ indestructible under $\kappa$-directed closed forcing*, Israel J. Math.**29**(1978), no. 4, 385–388. MR**472529**, DOI 10.1007/BF02761175 - Menachem Magidor,
*On the singular cardinals problem. I*, Israel J. Math.**28**(1977), no. 1-2, 1–31. MR**491183**, DOI 10.1007/BF02759779 - Alan H. Mekler,
*How to construct almost free groups*, Canadian J. Math.**32**(1980), no. 5, 1206–1228. MR**596105**, DOI 10.4153/CJM-1980-090-1 - Telis K. Menas,
*A combinatorial property of $p_{k}\lambda$*, J. Symbolic Logic**41**(1976), no. 1, 225–234. MR**409186**, DOI 10.2307/2272962 - E. C. Milner and S. Shelah,
*Some theorems on transversals*, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Vols. I, II, III, Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam, 1975, pp. 1115–1126. MR**0376358** - Saharon Shelah,
*A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals*, Israel J. Math.**21**(1975), no. 4, 319–349. MR**389579**, DOI 10.1007/BF02757993 - Saharon Shelah,
*On successors of singular cardinals*, Logic Colloquium ’78 (Mons, 1978) Studies in Logic and the Foundations of Mathematics, vol. 97, North-Holland, Amsterdam-New York, 1979, pp. 357–380. MR**567680** - Saharon Shelah,
*Proper forcing*, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR**675955** - Saharon Shelah,
*Incompactness in regular cardinals*, Notre Dame J. Formal Logic**26**(1985), no. 3, 195–228. MR**796637**, DOI 10.1305/ndjfl/1093870869
—, - Robert M. Solovay, William N. Reinhardt, and Akihiro Kanamori,
*Strong axioms of infinity and elementary embeddings*, Ann. Math. Logic**13**(1978), no. 1, 73–116. MR**482431**, DOI 10.1016/0003-4843(78)90031-1

*Applications of Radin’s forcing*(to appear).

*Cardinal arithmetics*, Oxford Univ. Press (to appear).

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**7**(1994), 769-830 - MSC: Primary 03E35; Secondary 03E55, 03E75, 20K27
- DOI: https://doi.org/10.1090/S0894-0347-1994-1249391-8
- MathSciNet review: 1249391