Suppose we have a first order theory $T$ in a language which contains the identity symbol $=$, constants $a_1$, $a_2$, $a_3$, ... $b_1$, $b_2$, $b_3$, ... and no relation or function symbols. For natural numbers $i \neq j$, suppose that $a_i \neq a_j$ is contained in $T$, and let us suppose that $T$ has no models whose domain consists only of $a_1, a_2, a_3, ...$ - that is to say, let us suppose that in every model of $T$, at least one of the $b$s is distinct from all of the $a$s.

Must there be some number $n$ such that in every model of $T$, one of $b_1$, $b_2$, ... , $b_n$ is distinct from all the $a$s?

Suppose we have a first order theory $T$ in a language which contains the identity symbol $=$, constants $a_1, a_2, a_3, \dotsc, b_1, b_2, b_3, \dotsc$ and no relation or function symbols. For natural numbers $i \neq j$, suppose that $a_i \neq a_j$ is contained in $T$, and let us suppose that $T$ has no models whose domain consists only of $a_1, a_2, a_3, \dotsc$ — that is to say, let us suppose that in every model of $T$, at least one of the $b$s is distinct from all of the $a$s.

Must there be some number $n$ such that in every model of $T$, one of $b_1, b_2, \dotsc, b_n$ is distinct from all the $a$s?