Sh:1019

• Shelah, S. Model theory for a compact cardinal. Israel J. Math. To appear. arXiv: 1303.5247
• Abstract:
  We would like to develop classification theory for $T$, a complete theory in $\mathbb{L} _{\theta,\theta}(\tau)$ when $\theta$ is a compact cardinal. We already have bare bones stability theory and it seemed we can go no further. Dealing with ultrapowers (and ultraproducts) naturally we restrict ourselves to “$D$ a $\theta$-complete ultrafilter on $I$, probably $(I,\theta)$-regular". The basic theorems of model theory work and can be generalized (like Łos’ theorem), but can we generalize deeper parts of model theory?

  The first section is trying to sort out what occurs to the notion of “stable $T$" for complete $\mathbb{L} _{\theta,\theta}$-theories $T$. We generalize several properties of complete first order $T$, equivalent to being stable (see the author classification theory book and find out which implications hold and which fail.

  In particular, can we generalize stability enough to generalize Chap.VI of the author classification theory book Let us concentrate on saturation in the local sense (types consisting of instances of one formula). We prove that at least we can characterize the $T$-s (of cardinality $\le \theta$ for simplicity) which are minimal for appropriate cardinal $\lambda \ge 2^\kappa + |T|$ in each of the following two senses. One is generalizing Keisler order $\triangleleft$, which measures how saturated ultrapowers are. Another generalizes the results on $\triangleleft^*$. That is, we ask: “Is there an $\mathbb{L} _{\theta,\theta}$-theory $T_1 \supseteq T$ of cardinality $|T| + 2^\theta$ such that for every model $M_1$ of $T_1$ of cardinality $> \lambda$, the $\tau(T)$-reduct $M$ of $M_1$ is $\lambda^+$-saturated?" Moreover, the two versions of stable used in the characterization are different.

• Version 2024-02-29 (59p)

@article{Sh:1019,
 author = {Shelah, Saharon},
 title = {{Model theory for a compact cardinal}},
 journal = {Israel J. Math.},
 year = {to appear},
 note = {\href{https://arxiv.org/abs/1303.5247}{arXiv: 1303.5247}},
 arxiv_number = {1303.5247}
}