It is well known that there are strong links between Set Theory and Topology/Real Analysis.
For instance, the study of Suslin's Problem turns out to be a set theoretic problem, even though it started in topology: namely, whether $\mathbb{R}$ is the only complete dense unbounded linearly ordered set that satisfies the c.c.c.
Another instance is when we see that what's behind extending Lebesgue Measure is really the theory of large cardinals, with the introduction of measurable cardinals. Also another example of a real analysis problem that ends up in Set Theory is whether every set of reals is measurable. So the links are clear between Set Theory and Topology/Real analysis.
My question is this: are there links, as strong as the ones I roughly described in the last paragraph, between Set Theory and Abstract Algebra? The only example I know of is the Set Theoretic solution to the famous Whitehead Problem by Shelah (namely that if $V=L$ then every Whitehead group is free and if MA+$\neg$CH then there is a Whitehead group which is not free).
Can we hope to discover more of these type of links between Set Theory and Abstract Algebra? In contrast, Model Theory seems to be strongly grounded in Abstract algebra. I have seen that Shelah has some papers about uncountable free Abelian groups but he seems to be the only one investigating some areas of Abstract Algebra with the help of Set Theory. So again is there hope for links?
Best Answer
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)."