# How to know how many models of certain size does a given theory has?

I'm stuck in this problem where i have a language $L$ with signature being a one equivalence relationship $R$, and a theory over this relation with infinite many infinite equivalence classes.

I need to know how many models of cardinality $\aleph_0$ are ther for this Theory, everyone around seem to find trivial to claim that there is just one model with this characteristics, but i don't get how to prove this. I've been trying to build an isomorphism between two arbitrary countable models $\mathfrak{A}$, $\mathfrak{B}$. Making $h(a) = b$ for some $a$ $\in A$ and $b \in B$ But i don't know how to keep going.

• If $M$ is a model of size $\aleph_0$ you can partition $M$ as $\displaystyle\bigsqcup_{n<\omega} M_n$ where each $M_n$ is an equivalence class, so an infinite (countable) subset. If $N$ is another countable model you can do the same; can you see the isomorphism then ? – Max Apr 4 '18 at 6:08
• so i can define the isomorphism based on an enumeration of each element from $M$ ,say $a_{i j}$ to denote the $j$ th element in $M_i$ equivalence class, to another element $b_{i j}$ in a $N_i$ equivalence class from the other model? – José Manuel Madrigal Ramírez Apr 4 '18 at 6:50
• Yes that's exactly it – Max Apr 4 '18 at 8:16