发表机构
University of Illinois Chicago（伊利诺伊大学芝加哥分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出无挠阿贝尔群Borel完备性的简化证明，并构造一个无计算副本但可从所有非计算集计算的群，解决Downey-Goncharov问题。
AI 中文摘要
最近，Paolini和Shelah证明了无挠阿贝尔群是Borel完备的，Laskowski和Ulrich给出了另一个证明。这意味着无挠阿贝尔群的同构问题具有尽可能高的复杂度，因此无挠阿贝尔群不存在任何有意义且完备的同构不变量。我们遵循Paolini-Shelah论证的思路给出一个证明，但在若干说明性方面具有优势。特别是，论证中的组合部分被大大简化。我们利用该证明的思想，证明存在一个无挠阿贝尔群，它没有可计算副本，但具有可从每个非可计算集合计算的副本。这回答了Downey和Goncharov的一个长期悬而未决的问题。
英文摘要
It was recently shown by Paolini and Shelah, with another proof by Laskowski and Ulrich, that torsion-free abelian groups are Borel complete. This means that isomorphism for torsion-free abelian groups is as complicated as possible, so that torsion-free abelian groups do not admit any meaningful complete isomorphism invariants. We give a proof following the ideas of the Paolini-Shelah argument but with several expository advantages. In particular, the combinatorial part of the argument is much simplified. We make use of the ideas of this proof to show that there is a torsion-free abelian group with no computable copy but with a copy computable from every non-computable set. This answers a long-standing question of Downey and Goncharov.
