$\leq_{SP}$ Can Have Infinitely Many Classes

TL;DR

This paper demonstrates that, under certain set-theoretic assumptions, the class of simple theories under Keisler's order can be infinitely partitioned, revealing new structural insights into model theory.

Contribution

It introduces the property of $ ext{≤}k$-type amalgamation for simple theories and constructs theories with specific amalgamation properties to show infinitely many $ ext{≤}_{SP}$-classes.

Findings

$ ext{≤}k$-type amalgamation distinguishes classes of simple theories.

Constructs theories $T_{n,k}$ with specific amalgamation properties.

Shows the maximal $ ext{≤}_{SP}$-class is the class of unsimple theories.

Abstract

Building off of recent results on Keisler's order, we show that consistently, $\leq_{SP}$ has infinitely many classes. In particular, we define the property of $\leq k$-type amalgamation for simple theories, for each $2 \leq k < \omega$. If we let $T_{n, k}$ be the theory of the random $k$-ary, $n$-clique free random hyper-graph, then $T_{n, k}$ has $\leq k-1$-type amalgamation but not $\leq k$-type amalgamation. We show that consistently, if $T$ has $\leq k$-type amalgamation then $T_{k+1, k} \not \leq_{SP} T$, thus producing infinitely many $\leq_{SP}$-classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal $\leq_{SP}$-class is exactly the class of unsimple theories. Finally, we show that consistently, if $T$ has $<\aleph_0$-type amalgamation, then $T \leq_{SP} T_{rg}$, the theory of the random graph.