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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?

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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)."

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Simon, are you really using the full Ramsey cardinal, or is this something like 0-sharp, or less? Is this property known to have a large cardinal lower bound? –  Joel David Hamkins Apr 7 '10 at 1:15
I use $\Sigma^{1}_{3}$ absoluteness for notions of forcing which collapse the continuum to a countable set ... so a Ramsey is a reasonable assumption even if it isn't optimal. The result almost certainly doesn't need large cardinals for its consistency. I would guess it is true if you add lots of Cohen reals. However, I prefer to use the existence of a "small" large cardinal as this shows that the result is true in the actual set-theoretic universe. –  Simon Thomas Apr 7 '10 at 1:53

A curious example is the linear ordering on braid groups, first discovered by Patrick Dehornoy as a consequence of a large cardinal axiom. Proofs without set theory were discovered later -- and also earlier, but unpublished, by Thurston -- but Dehornoy believes that intuition from set theory was crucial in his discovery of the result. See his book Braids and Self Distributivity, particularly the Introduction, which is available here.

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This work is connected with Richard Laver's work on free left-distributive algebras and Laver tables. The large cardinals enter into it in Laver observation that if j:V_lambda to V_lambda witnesses the I3 large cardinal axiom, then the algebra generated by j under composition and application h(k) = union_alpha h(k|V_alpha) is a free left distributive algebra. There were no other natural presentations of this algebra (except for the formal term algebra), and it was by using the large cardinal properties of this context that Laver, Dehornoy and others could gain insight into the free algebra. –  Joel David Hamkins Apr 7 '10 at 12:28
It is also interesting to follow up Thurston's idea, which is expounded by Short & Wiest, L'Enseignement Mathematique 46(2000) pp. 279--312 (available online). It turns out to follow quite naturally from the idea of Jakob Nielsen (1927) to represent mappings of surfaces by mappings of the "boundary at infinity" of the universal cover, which is a circle (or a line, if you take the half-plane model of the hyperbolic plane ). This is where the linear order comes from. Knowing this, I am even more amazed that there is a route to the same theorem via large cardinals. –  John Stillwell Apr 8 '10 at 0:12

Let me also mention some of the work that's been done on the automorphism tower problem in group theory. This topic was the mentioned in this early classic MO question (from before my time here). The basic situation is that one starts with a group G, and then iteratively computes the automorphism group Aut(G), the automorphism group of THAT group, and so on. The automorphism tower can be continued transfinitely, by taking a direct limit of the natural system of homomorphisms mapping a group element to the corresponding inner automorphism. When the group G is centerless, then every group in the tower is centerless, and the groups can be viewed as growing. The main question is: does the tower ever stop growing? Does one reach a fixed point? Simon Thomas proved that the answer is yes for centerless groups and I proved yes for all groups, by showing that every group leads eventually to a centerless group.

But the connections with set theory become very interesting. Simon Thomas and I proved that there can be a group G, whose automorphism tower depends highly on the set-theoretic background, in the sense that there are forcing extensions of the universe in which the very same group G has towers of different height. We can make the tower taller or shorter, as desired. The point is that even if one has the same group G, then the the automorphism group Aut(G) already depends on the set-theoretic background, since one can sometimes add new generic automorphisms by forcing. For example, it is sometimes possible for a complete group (centerless + no outer automorphisms) to gain new outer automorphisms in a forcing extension. This would be an example of a tower increasing from height 0 to height at least 1 (and it might still grow much taller!). This phenomenon has now been extended by Gunter Fuchs and Philipp Lücke, who showed that almost any successive up-down pattern is achievable in subsequent forcing extensions, by iterated forcing. The general conclusion is that the automorphism tower of a group, even a finite group, exhibits a fundamentally set-theoretic nature, akin to iteratively computing the power set.

Simon Thomas, who I see has posted an answer to this question, is currently writing a book on the automorphism tower problem, and it is excellent, from the preliminary versions I've seen.

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Thank you, that is exactly what I was looking for. I did not know of these deep connections. This sounds fascinating. –  Carlo Von Schnitzel Apr 7 '10 at 1:17
Is MO really old enough to have classical and modern eras? –  Qiaochu Yuan Apr 7 '10 at 1:49
Qiaochu, of course, I made a joke. But anyway that question was the reason I first came to MO, because a colleague (Kevin O'Bryant) had noticed it and dropped by my office telling me about it. –  Joel David Hamkins Apr 7 '10 at 2:05

The following older MO question is pretty relevant: What are some reasonable-sounding statements that are independent of ZFC?

In particular, the highest voted answer (by Daniel Erman) is truly mindboggling: Here's an example from commutative algebra. The projective dimension of a module M is defined as the minimal length of a projective resolution of M. Let S be the ring ℂ[x,y,z] and M be the module ℂ(x,y,z). Then the projective dimension of M is undecidable in ZFC. More specifically, the projective dimension of M is 2 if the continuum hypothesis holds, and it is 3 if the continuum hypothesis fails.

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It turns out that when it comes to infinite groups/modules, some algebraic concepts are deeply connected to the underlying set theory (for example, the notion of freeness, the structure of Ext, etc). A good reference for this subject is the book "Almost free modules" by Eklof and Mekler. This book introduces the works of Shelah, Gobel, Eklof and many other important contributors in this field. This research has also led to some interesting developments in "pure" set theory, such as the introduction of black-boxes by Shelah (some diamond-like combinatorial principles which can be proved in ZFC alone, and allow the construction of many interesting algebraic objects).

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Shelah is definitely not alone! Here are few set theorists who have done substantial algebraic work.

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This is true, I am sorry I forgot them, they are definitely working in this area. Do you know if there are others one working in this area? –  Carlo Von Schnitzel Apr 7 '10 at 0:35
The above list is far from complete! (Apologies to the many that I didn't mention.) –  François G. Dorais Apr 7 '10 at 0:39
This is more of a question than a correction, but would one really class Vladimir Pestov as a set theorist? (I was under the impression he was a functional analyst, although I've never asked him about this.) –  Yemon Choi Apr 7 '10 at 0:41
Also, are there other results in Group Theory which are independent or which have a purely set theoretic setting? By the way I am actually looking for problems that have exclusively a set theoretic setting (if they exist) and not just examples of "applied set theory" to algebra, like Shelah says. –  Carlo Von Schnitzel Apr 7 '10 at 0:43
It has been known for set theorists to show that published "theorems" in group theory are actually independent of ZFC ... –  Simon Thomas Apr 7 '10 at 0:58

The subject of Borel Equivalence relation theory involves deep connections between set theory, particularly descriptive set theory, and classification problems in algebra. The principal theme of the subject is to investigate the complexity of various naturally-occuring equivalence relations, such as the isomorphism relation on finitely generated groups, which arise in other areas of mathematics. It turns out that many of these relations can be viewed as Borel relations on a standard Borel space, and they fit into a hierarchy under the concept of Borel reducibility, introduced by Harvey Friedman. I explained a little about the subject in this MO answer.

Much of the best work in this subject is characterized by deep connections between set theory and algebra.

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You may want to determine which discipline of set theory to link to another discipline. Descriptive Set Theory came (roughly) as a result of foundational issues arising from looking at certain arguments in Topology and Real Analysis, and at some point later ties to Model Theory, Proof Theory, and Recursion Theory were also investigated.

If you look at results in Universal Algebra, you will find many links to various foundational disciplines. This is probably the easiest source to find the kinds of links you mention. For example, as a weak parallel to Shelah's Classification Theory, one finds looking at varieties of algebras and considering their spectra, and classifying those which have many models in algebraic terms to those which have few models. People such as Baldwin, Jeong, Jezek, Kearnes, McKenzie, Valeriote, and Wood do work on decidability, the lattice of interpretability types, spectra, and other questions in the context of varieties.

There is also algebraic logic, cylindric algebras, and algebras being used to study certain aspects of set theory and logic. You may want to peruse some of that material and then revisit the question of what links you would still like to see.

Gerhard "Ask Me About System Design" Paseman, 2010.04.06

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Thank you for this suggestion, it helped me make my question precise: what I am hoping to see is problems that start in, say, Group Theory but end up creating a branch of Set Theory (if any exists) the same way asking in real analysis if every set of reals is measurable leads to purely Set Theoretic problems. Are there any such questions in Abstract Algebra that gets reformulated with the help of infinitary combinatorics or forcing and then "find their place there", so to speak. –  Carlo Von Schnitzel Apr 7 '10 at 1:02
I am glad I could help. I hope the other posters can steer you to exactly the link you want. Gerhard "Ask Me About System Design" Paseman, 2010.04.06 –  Gerhard Paseman Apr 7 '10 at 1:19

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