Greets
I just want to see examples of vaughtian pairs.
Thanks
Greets
I just want to see examples of vaughtian pairs.
Thanks
I think more generally Vaughtian pairs may be "constructed" as follows:
Let $\mathcal{N}$ is a model of uncountable cardinality $\kappa$ in some denumerable (relational) language. Suppose $\bar{b}$ is some tuple of elements of $N$ and $\varphi ( x , \bar{y} )$ is a formula such that $A = \{ x \in N : \mathcal{N} \models \varphi ( x , \bar{b} ) \}$ is infinite of cardinality $\lambda < \kappa$. By Downward Löwenheim-Skolem let $\mathcal{M}$ be an elementary submodel of $\mathcal{N}$ of cardinality $\lambda$ including $A \cup \{ \bar{b} \}$. Then $\mathcal{M} , \mathcal{N}$ will be a Vaughtian pair.