Torsion-free abelian groups are consistently $ {\rm a}\Delta ^1_2$-complete

Saharon Shelah; Douglas Ulrich

doi:10.4064/fm673-12-2018

Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

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Torsion-free abelian groups are consistently ${\rm a}\Delta ^1_2$ -complete

Volume 247 / 2019

Saharon Shelah, Douglas Ulrich Fundamenta Mathematicae 247 (2019), 275-297 MSC: Primary 03C55. DOI: 10.4064/fm673-12-2018 Published online: 12 August 2019

Abstract

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

Authors

Saharon ShelahInstitute of Mathematics
Hebrew University
Jerusalem, Israel
e-mail
Douglas UlrichDepartment of Mathematics
University of California, Irvine
Irvine, CA 92697, U.S.A.
e-mail

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Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Fundamenta Mathematicae / All issues

Fundamenta Mathematicae

Torsion-free abelian groups are consistently {\rm a}\Delta ^1_2-complete

Volume 247 / 2019

Abstract

Authors

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Torsion-free abelian groups are consistently ${\rm a}\Delta ^1_2$ -complete