I am studying the theory of independent unary predicates in a model theory textbook, and I am struggling to solve a problem that is left open in the notes. By the theory of independent unary predicates I mean the theory $T_n$ in language $L_n$ with $n$ unary disjoint predicates, each $P_i$ is satisfied by infinite number of elements from the domain, and also there are infinitely many elements in the domain that do not satisfy each $P_i$. Each $P_i$ is disjoint from all other $P_j$'s. I have shown that $T_n$ is $\aleph_0$-categorical, and from here, using the Los-Vaught Test, concluded that $T_n$ is complete. I now want to answer the following question: How many models of cardinality $\aleph_1$(up to isomorphism) does $T_2$ have?
I am really stuck on where to start with this, and any hints and help would be really appreciated. Thank you.
The theory you are describing is not the theory of independent unary predicates, but rather, the theory of disjoint unary predicates (although it does not change the properties of the theory very much, as long as there are only finitely many of them).
To find the number of models, show that if $M, N$ are models of $T$, then they are isomorphic if and only if all the atoms in the Boolean algebra generated by the predicates have the same number of elements in $M$ and $N$.
For $n$ disjoint infinite predicates with infinite complement, this leads to the conclusion that there are $2^{n+1}$ models of cardinality $\aleph_1$.