Shelah's proof of the independence of the Whitehead Problem

Question

In the beginning Shelah classifies all $\aleph_1$-free Abelian groups into 3 possibilities each of which is satisfied by some $\aleph_1$-free Abelian group and the classification depends on the group G up to isomorphism (in page 250 is the Israel Journal of Mathematics for those who have the article around). He also defines pure subgroups.

What I want to ask is : why are pure subgroups so important for the proof? It seems like decomposition of pure subgroups of groups is central in the V=L chunk of the proof and groups of possibility 1 and 2 are defined in terms of how well can they split as a direct sum of pure subgroups. Is it just because they are of prime importance for Abelian groups in general.

According to V=L, possibility 1 and 2 contradict Whiteheadness but with MA possibility 2 does not contradict Whiteheadness.

I hope my question is precise.

Thx

Answer 1

A good source for understanding Shelah’s proof of the undecidability of Whitehead’s problem is Paul Eklof’s “Whitehead’s Problem is undecidable” article in the American Mathematical Monthly. The key property about pure subgroups relevant to Shelah’s proof is that a countable torsion-free group is free iff every finitely generated subgroup is contained in a finitely generated pure subgroup. This gives a characterization of countable free groups is terms of objects that are “almost finite”. This also gives a decomposition of a countable free group into an increasing chain of finitely generated pure subgroups. Recall, that a group is torsion-free iff every finitely generated subgroup is free. This is generalized by the notion of $\aleph_1$-free which states that every countable subgroup is free. Pure subgroups are generalized to $\aleph_1$-pure subgroups where the quotient is correspondingly $\aleph_1$-free. The characterization of free groups of cardinality $\aleph_1$ with the generalized notion is not so simple. For a group of size $\aleph_1$ to be free it is not sufficient that every countable subgroup be contained in a countable $\aleph_1$-pure subgroup. However any group having the above property can be decomposed into an $\aleph_1$ chain of free subgroups with the subgroups indexed by the successor ordinals being $\aleph_1$-pure. If furthermore “enough” of the limit stages are also $\aleph_1$-pure, more precisely if the set of ordinals indexing $\aleph_1$-pure subgroups is stationary, then the group is free. Again this gives a characterization of free groups of size $\aleph_1$ in terms of smaller, namely countable subgroups. The $V=L$ case uses the fact that the $\Diamond$-principle holds in $L$. The $\Diamond$-principle allows you to anticipate properties of an object of size $\aleph_1$ as it is constructed from countable objects indexed by countable ordinals. This is where the characterization of a free group of size $\aleph_1$ in terms of its countable subgroups is crucial. I believe the fact that pure subgroups are “almost finite” allows Shelah to prove that a certain poset has the countable chain condition in the Martin’s axiom part of the proof.