# Sh:1141

- Shelah, S., & Ulrich, D. (2019).
*Torsion-free abelian groups are consistently {\mathrm{a}}\Delta^1_2-complete*. Fund. Math.,**247**(3), 275–297. arXiv: 1804.08152 DOI: 10.4064/fm673-12-2018 MR: 4017015 -
Abstract:

Let \textrm{TFAG} be the theory of torsion-free abelian groups. We show that if there is no countable transitive model of ZF^- + \kappa(\omega) exists, then \textrm{TFAG} is a \Delta^1_2-complete; in particular, this is consistent with ZFC. We define the \alpha-ary Schröder- Bernstein property, and show that \textrm{TFAG} fails the \alpha-ary Schröder-Bernstein property for every \alpha < \kappa(\omega). We leave open whether or not \textrm{TFAG} can have the \kappa(\omega)-ary Schröder-Bernstein property; if it did, then it would not be a \Delta^1_2-complete, and hence not Borel complete. - Version 2018-04-25_11 (21p) published version (24p)

Bib entry

@article{Sh:1141, author = {Shelah, Saharon and Ulrich, Douglas}, title = {{Torsion-free abelian groups are consistently ${\mathrm{a}}\Delta^1_2$-complete}}, journal = {Fund. Math.}, fjournal = {Fundamenta Mathematicae}, volume = {247}, number = {3}, year = {2019}, pages = {275--297}, issn = {0016-2736}, mrnumber = {4017015}, mrclass = {03C55 (20K15 20K20)}, doi = {10.4064/fm673-12-2018}, note = {\href{https://arxiv.org/abs/1804.08152}{arXiv: 1804.08152}}, arxiv_number = {1804.08152} }