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The isomorphism relation on countable torsion free abelian groups. (English) Zbl 1021.03042
It is proved that there is no reasonable scheme for classifying countable torsion-free abelian groups. More precisely, it is proved that every every Borel isomorphism relation on a class of countable structures can be embedded, in a Borel way, into the isomorphism relation of countable torsion-free abelian groups, which we denote $$\cong |_{\text{TFA}}$$; it follows, by a result of Harrington, that $$\cong |_{\text{TFA}}$$ is not Borel. This is accomplished by a careful coding technique. Familiarity is presumed with a previous comprehensive study of Borel equivalence relations by the author, A. S. Kechris and A. Louveau [Ann. Pure Appl. Logic 92, 63-112 (1998; Zbl 0930.03058)]. This paper also corrects an earlier claim by the author [in: P. C. Eklof et al. (eds.), Abelian groups and modules. Proceedings of the international conference, Dublin, 1998, 269-292 (1999; Zbl 0983.20053)] that the isomorphism relation for countable structures over any countable language can be embedded in $$\cong |_{\text{TFA}}$$; this question remains open.

##### MSC:
 03E15 Descriptive set theory 20K20 Torsion-free groups, infinite rank
##### Keywords:
Borel equivalence relation; torsion-free abelian group
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