# 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.
• 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.
• Current version: 2018-04-25_10 (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}
}