Disjoint $n$-amalgamation and pseudofinite countably categorical theories

Abstract

Disjoint $n$-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this paper, we show that if a countably categorical theory $T$ admits an expansion with disjoint $n$-amalgamation for all $n$, then $T$ is pseudofinite. All theories which admit an expansion with disjoint $n$-amalgamation for all $n$ are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain non-simple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T_{\text{feq}}^{\ast }$ and $T_{\text{CPZ}}$, and show that both are pseudofinite. The theories $T_{\text{feq}}^{\ast }$ and $T_{\text{CPZ}}$ are not simple, but they are NSOP${}_1$. This is established here for $T_{\text{CPZ}}$ for the first time.

revised version, to appear in Notre Dame Journal of Formal Logic

References in corpus (2)

• Simple structures axiomatized by almost sure theories
• The infinite random simplicial complex

Cited by in corpus (3)

• The Generic Bipartite Graphs of Diameter 3: Their Ages and their Almost Sure Theories
• Classification properties for some ternary structures
• MS-measurability via Coordinatization