Vaught's Conjecture for Almost Chainable Theories

Abstract

A structure $\mathbb{Y}$ of a relational language $L$ is called almost chainable iff there are a finite set $F\subset Y$ and a linear order $<$ on the set $Y\setminus F$ such that for each partial automorphism $\phi$ (i.e., local automorphism, in Fra\"ıssé's terminology) of the linear order $⟨Y\setminus F,<⟩$ the mapping ${\mathrm{id}}_F\cup \phi$ is a partial automorphism of $\mathbb{Y}$. By a theorem of Fra\"ıssé, if $|L|<\omega$, then $\mathbb{Y}$ is almost chainable iff the profile of $\mathbb{Y}$ is bounded; namely, iff there is a positive integer $m$ such that $\mathbb{Y}$ has $\le m$ non-isomorphic substructures of size $n$, for each positive integer $n$. A complete first order $L$-theory $\mathcal{T}$ having infinite models is called almost chainable iff all models of $\mathcal{T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of $\mathcal{T}$. In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

16 pages, 1 figure