I'm working on a problem from Kees Doets, and he mentions the following structures:
$X=(Q,<,n)_{n \in N}$
$Y=(Q,<,\frac{-1}{n+1})_{n \in N}$
$Z=(Q,<,q_n)_{n \in N}$ where $\{q_n\}_{n \in N}$ is an ascending sequence of rationals converging to some irrational.
The problem asks to show that up to isomorphism $Y$ and $Z$ are the only other countable models of $Th(X)$. Also asks to show which one is saturated and which one is prime. But for now I have a much more basic question:
What does a sentence in $Th(X)$ look like? I am confused by the symbols $n \in N$ added to the language. Can a sentence $\phi \in Th(X)$ for example be $\phi=\forall q \in Q, \exists n \in N: q<n$
This is true in the rationals, but is a sentence allowed to quantify over the constant symbols in its language, or is quantification reserved only for variables which will be interpreted in the model? Assuming of course that the interpretation of the symbols $n$ in the model is actually the natural numbers.