On 0, 1-laws and asymptotics of definable sets in geometric Fraïssé classes
Volume 239 / 2017
Fundamenta Mathematicae 239 (2017), 201-219
MSC: 03C13, 03C15, 03C45, 05A16.
DOI: 10.4064/fm122-1-2017
Published online: 23 June 2017
Abstract
We examine one consequence for the generic theory $T_\mathbf {C}$ of a geometric Fraïssé class $\mathbf {C}$ when $\mathbf {C}$ has the $0,1$-law for first-order logic with convergence to $T_\mathbf {C}$ itself. We show that in this scenario, if the asymptotic probability measure in play is not terribly exotic, then $\mathbf {C}$ is “very close” to being a 1-dimensional asymptotic class—so that $T_\mathbf {C}$ is supersimple of finite $SU$-rank.
