Sh:977

• Shelah, S. (2012). Modules and infinitary logics. In Groups and model theory, Vol. 576, Amer. Math. Soc., Providence, RI, pp. 305–316. arXiv: 1011.3581 DOI: 10.1090/conm/576/11376 MR: 2962893
• Abstract:
  We deal with Abelian groups and R-modules. We consider theories in infinitary logic of the form \mathbb{L}_{\lambda,\theta} of such structures M and prove they have elimination of quantifiers up to positive existential formulas, (so ones defining subgroups of some power of M). However, we demand that we expand by enough individual constants. Hence those theories are stable in the appropriate sense and understood to some extent.

  In 2026, John Baldwin pointed out a mistake in the end of the proof of the main claim of Section 4, which is used in the theorem in Section 2, and made further requests for clarifications. Here this is corrected, in addition to other improvements. The error is corrected in three ways — in Section 2 we can use a weaker version of Section 4, and in section 4 we also get the original result with more assumptions on the cardinal. Lastly, we provide a shorter and self-contained proof of the main theorem in §2. We can use the older version of Section 4, but then we use somewhat larger cardinals.

• Version 2026-06-09 (23p) published version (12p)

@incollection{Sh:977,
 author = {Shelah, Saharon},
 title = {{Modules and infinitary logics}},
 booktitle = {{Groups and model theory}},
 series = {Contemp. Math.},
 volume = {576},
 year = {2012},
 pages = {305--316},
 publisher = {Amer. Math. Soc., Providence, RI},
 mrnumber = {2962893},
 mrclass = {03C60 (03C10 03C45 03C75 16B70)},
 doi = {10.1090/conm/576/11376},
 note = {\href{https://arxiv.org/abs/1011.3581}{arXiv: 1011.3581}},
 arxiv_number = {1011.3581}
}