Descriptive set theory also has something to say about algebra ... For example, the Higman-Neumann-Neumann Embedding Theorem states that any countable group G can be embedded into a 2-generator group K. In the standard proof of this classical theorem, the construction of the group K involves an enumeration of a set of generators of the group G; and it is clear that the isomorphism type of K usually depends upon both the generating set and the particular enumeration that is used. So it is natural to ask whether there is a more uniform construction with the property that the isomorphism type of K only depends upon the isomorphism type of G. As if ...
Assume the existence of a Ramsey cardinal and suppose that G |----> F(G) is a Borel map from the space of countable groups to the space of finitely generated groups such that G embeds into F(G). Then there exists an uncountable set of pairwise isomorphic groups G such that the f.g. groups F(G) are pairwise incomparable with respect to relative constructibility; ie while G, H are isomorphic, F(G) doesn't even lie in the "set-theoretic universe generated by F(H)."
