≤_SP Can Have Infinitely Many Classes

2018/04/23 by Saharon Shelah, Shelah, Saharon, Ulrich, Danielle
Computer Science · Mathematics · #03C55 #03E35 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.1804.08523

openalex publication_date 2018/04/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Building off of recent results on Keisler's order, we show that consistently, ≤_SP has infinitely many classes. In particular, we define the property of ≤ k-type amalgamation for simple theories, for each 2 ≤ k < ω. 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 ≤ k-1-type amalgamation but not ≤ k-type amalgamation. We show that consistently, if T has ≤ k-type amalgamation then T_{k+1, k} \not ≤_SP T, thus producing infinitely many ≤_SP-classes. The same construction gives a simplified proof of Shelah's theorem that consistently, the maximal ≤_SP-class is exactly the class of unsimple theories. Finally, we show that consistently, if T has <ℵ₀-type amalgamation, then T ≤_SP T_rg, the theory of the random graph.

Related

• Amalgamation and Keisler's Order
• An example of a new simple theory
• Keisler's order has infinitely many classes
• Keisler's order is not simple (and simple theories may not be either)
• Some simple theories from a Boolean algebra point of view