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Rachid Atmai
Rachid Atmai
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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

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 the V=L chunk of the proof 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

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

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Rachid Atmai
Rachid Atmai
  • 3.8k
  • 2
  • 24
  • 36

Shelah's proof of the independence of the Whitehead Problem

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 the V=L chunk of the proof 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