Examples of vaughtian pairs

Question

Greets

I just want to see examples of vaughtian pairs.

Thanks

Answer 1

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.

• The example in the answer above is certainly of this type.
• Consider an uncountable linear order $\mathcal{N} = ( N , < )$ with an element $b \in N$ such that $A = \{ x \in N : x < b \}$ is countably infinite. Letting $\mathcal{M}$ be a countable elementary submodel of $\mathcal{N}$ including $A \cup \{ b \}$, and then $\mathcal{M} , \mathcal{N}$ will be as required.