TY - JOUR
T1 - When does almost free imply free? (for groups, transversals, etc.)
AU - Magidor, Menachem
AU - Shelah, Saharon
PY - 1994/10
Y1 - 1994/10
N2 - We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal κ to ℵκ+1. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for “κ 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 (Euqation presented) free implies (Euqation presented) free. Similar construction yields a model in which ℵκ free implies free for k the first cardinal fixed point (namely, the first cardinal α satisfying α = ℵα). 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.
AB - We show that the construction of an almost free nonfree Abelian group can be pushed from a regular cardinal κ to ℵκ+1. Hence there are unboundedly many almost free nonfree Abelian groups below the first cardinal fixed point. We give a sufficient condition for “κ 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 (Euqation presented) free implies (Euqation presented) free. Similar construction yields a model in which ℵκ free implies free for k the first cardinal fixed point (namely, the first cardinal α satisfying α = ℵα). 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.
UR - http://www.scopus.com/inward/record.url?scp=84968510169&partnerID=8YFLogxK
U2 - 10.1090/S0894-0347-1994-1249391-8
DO - 10.1090/S0894-0347-1994-1249391-8
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AN - SCOPUS:84968510169
SN - 0894-0347
VL - 7
SP - 769
EP - 830
JO - Journal of the American Mathematical Society
JF - Journal of the American Mathematical Society
IS - 4
ER -