# Sh:1019

• Shelah, S. Model theory for a compact cardinal. Preprint. arXiv: 1303.5247
• Abstract:
We like to develop model theory for T, a complete theory in \mathbb{L}_{\theta,\theta}(\tau) when \theta is a compact cardinal. By [Sh:300a] we have bare bones stability and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) we restrict ourselves to “D a \theta-complete ultrafilter on I, probably (I,\theta)-regular". The basic theorems work, but can we generalize deeper parts of model theory? In particular can we generalize stability enough to generalize [Sh:c, Ch.VI]? We prove that at least we can characterize the T’s which are minimal under Keisler’s order, i.e. such that \{D:D is a regular ultrafilter on \lambda and M \models T \Rightarrow M^\lambda/D is \lambda-saturated\}. Further we succeed to connect our investigation with the logic \mathbb{L}^1_{< \theta} introduced in [Sh:797]: 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. 2013.11.14 Doron will read it for a grade, have to finish till 2013.12.31. Updates 2013.12.27 Have update, revise §2 strongly now have both versions of minimality- different characterization;
• Version 2019-03-13_13 (59p)
Bib entry
@article{Sh:1019,
author = {Shelah, Saharon},
title = {{Model theory for a compact cardinal}},
note = {\href{https://arxiv.org/abs/1303.5247}{arXiv: 1303.5247}},
arxiv_number = {1303.5247}
}