Logic Seminar

In this talk we shall discuss some anticlassification results for orderable groups. First, we introduce the space of Archimedean orderings Ar(G)Ar⁡(G) for a given countable orderable group GG. We prove that the equivalence relation induced by the natural action of GL2(Q)GL2⁡(Q) on Ar(Q2)Ar⁡(Q2) is not concretely classifiable. Then we shall discuss the complexity of the isomorphism relation for countable ordered Archimedean groups. In particular, we show that its potential class is not Π03Π30. This topological constraint prevents classifying ordered Archimedean groups using countable subsets of reals. Our proofs combine classical results on Archimedean groups, the theory of Borel equivalence relations, and analyzing definable sets in the basic Cohen model and other models of Zermelo-Fraenkel set theory without choice. This is joint work with Dave Marker, Luca Motto Ros, and Assaf Shani.