Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: It is consistent with ZFC that B_1-groups are not B_2-groups   A torsion-free abelian group B of arbitrary rank is called a B_1-group if
Bext^1(B,T)=0 for every torsion abelian group T, where Bext^1 denotes the group
of equivalence classes of all balanced exact extensions of T by B. It is a
long-standing problem whether or not the class of B_1-groups coincides with the
class of B_2-groups. A torsion-free abelian group B is called a B_2-group if
there exists a continuous well-ordered ascending chain of pure subgroups,
  0=B_0 subset B_1 subset ... subset B_alpha subset ... subset B_lambda
=B=bigcup_{alpha in lambda} B_alpha such that B_{alpha+1}=B_alpha+G_alpha for
every alpha in lambda for some finite rank Butler group G_alpha. Both,
B_1-groups and B_2-groups are natural generalizations of finite rank Butler
groups to the infinite rank case and it is known that every B_2-group is a
B_1-group. Moreover, assuming V=L it was proven that the two classes coincide.
Here we demonstrate that it is undecidable in ZFC whether or not all B_1-groups
are B_2-groups. Using Cohen forcing we prove that there is a model of ZFC in
which there exists a B_1-group that is not a B_2-group.

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The categories of this paper are: math-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: econ
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: gr-qc
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an