Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete

Saharon Shelah; Douglas Ulrich

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arXiv · 1804.08152

Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete

Saharon Shelah·Douglas Ulrich

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Abstract

Let $\mbox{TFAG}$ be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of $ZFC^- + κ(ω)$ exists, then $\mbox{TFAG}$ is $a Δ^1_2$-complete; in particular, this is consistent with $ZFC$. We define the $α$-ary Schröder- Bernstein property, and show that $\mbox{TFAG}$ fails the $α$-ary Schröder-Bernstein property for every $α< κ(ω)$. We leave open whether or not $\mbox{TFAG}$ can have the $κ(ω)$-ary Schröder-Bernstein property; if it did, then it would not be $a Δ^1_2$-complete, and hence not Borel complete.

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BibTeX RIS

Saharon Shelah, Douglas Ulrich. 2018-04-22. Torsion-Free Abelian Groups are Consistently $a Δ^1_2$-complete. https://arxiv.org/abs/1804.08152

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