Keisler's Order

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A theory is complete when it decides every sentence in its first-order language, and countable when its language is countable; theories here are consistent. An ultrafilter $\mathcal D$ on a set $I$ is a family of subsets closed under supersets and finite intersections, omitting $\varnothing$, and containing exactly one of $X,I\setminus X$ for each $X\subseteq I$. For $|I|=\lambda$ infinite, it is regular if there are $X_\alpha\in\mathcal D$ for $\alpha<\lambda$ such that each $i\in I$ belongs to only finitely many $X_\alpha$. The ultrapower $M^I/\mathcal D$ consists of functions $I\to M$ modulo equality on a set in $\mathcal D$; operations and relations are evaluated coordinatewise modulo $\mathcal D$. A model is $\lambda^+$-saturated if every complete one-variable type over any parameter set of cardinality at most $\lambda$, consistent with its elementary diagram, is realized in the model. A type is a consistent set of formulas choosing each formula or its negation.

For complete countable theories $T_1,T_2$, set $T_1\unlhd T_2$ when, for every infinite cardinal $\lambda$, every regular ultrafilter $\mathcal D$ on a set $I$ of size $\lambda$, and every $M_j\models T_j$, \[ M_2^I/\mathcal D\text{ is }\lambda^+\text{-saturated} \quad\Longrightarrow\quad M_1^I/\mathcal D\text{ is }\lambda^+\text{-saturated}. \] Here $\lambda^+$ is the least cardinal greater than $\lambda$. Define $T_1\sim T_2$ if both $T_1\unlhd T_2$ and $T_2\unlhd T_1$ hold. The resulting order on equivalence classes is Keisler's order.

Statement recorded in the supplied source.

Determine the complete structure of Keisler's order: classify its equivalence classes of complete countable theories and determine all comparisons $[T_1]\unlhd[T_2]$ between them. A description only of the stable part or a finite list of known classes does not give the requested global classification. [S2]

Short English statement

Classify first-order theories according to which regular-ultrafilter ultrapowers become saturated, and determine every comparison between the resulting classes.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1467 (MOD-010). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Maryanthe Malliaris and Saharon Shelah, Keisler’s Order Has Infinitely Many Classes, arXiv:1503.08341v2 (2015), introduction and Section 6, Definition 6.4. https://arxiv.org/pdf/1503.08341v2.