# Sh:1101

• Shelah, S. (2021). Isomorphic limit ultrapowers for infinitary logic. Israel J. Math., 246(1), 21–46.
• Abstract:
The logic \mathbb{L} ^1_\theta was introduced in [Sh:797]; it is the maximal logic below \mathbb{L} _{\theta,\theta} in which a well ordering is not definable. We investigate it for \theta a compact cardinal. We prove that it satisfies several parallel of classical theorems on first order logic, strengthening the thesis that it is a natural logic. In particular, two models are \mathbb{L} ^1_\theta-equivalent iff for some \omega-sequence of \theta-complete ultrafilters, the iterated ultra-powers by it of those two models are isomorphic.

Also for strong limit \lambda > \theta of cofinality \aleph_0, every complete \mathbb{L} ^1_\theta-theory has a so called a special model of cardinality \lambda, a parallel of saturated. For first order theory T and singular strong limit cardinal \lambda , T has a so called special model of cardinality \lambda. Using “special" in our context is justified by: it is unique (fixing T and \lambda), all reducts of a special model are special too, so we have another proof of interpolation in this case.

This was earlier section 3 of [Sh:1019].

• Version 2021-09-13_5 (21p) published version (26p)
Bib entry
@article{Sh:1101,
author = {Shelah, Saharon},
title = {{Isomorphic limit ultrapowers for infinitary logic}},
journal = {Israel J. Math.},
fjournal = {Israel Journal of Mathematics},
volume = {246},
number = {1},
year = {2021},
pages = {21--46},
issn = {0021-2172},
mrnumber = {4358271},
mrclass = {03C45 (03C30 03C55)},
doi = {10.1007/s11856-021-2226-x},
note = {\href{https://arxiv.org/abs/1810.12729}{arXiv: 1810.12729}},
arxiv_number = {1810.12729}
}