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 an 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, the referee for a paper with Paolini 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. Here this is corrected and more. One change is that we now prove the result in Section 4, but using a somewhat larger cardinal (bigger than the original — not in the sense of being a large cardinal), so the old proof in Section 2 works. The proof in Section 2 has also been made more economical.

• Version 2026-05-01_3 (19p) 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}
}