Setting
Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. So we see $T$ has countably many axioms. How many models of $T$ are there of cardinality $\aleph_1$,$\aleph_2$,$\aleph_{\omega_1}$?
Problem
First, I'd like to clarify that by "model $\mathcal{M}$ of cardinality k", it means the universe underlying $\mathcal{M}$ has cardinality $k$, correct?
My next confusion lies in in cardinal arithmetic more than anything else. I can see $T$ admits only one model of cardinality $\aleph_0$ up to isomorphism. But for $\aleph_1$ I do not know how to "count" the number of models. My argument is below.
so we have $$\text{cardinality of each model} \times \text{cardinality of number of models} = \aleph_1.$$ This relation maybe satisfied if \begin{align*} &(\text{cardinality of each model} = \aleph_1) ~\vee~ (\text{cardinality of number of models} = \aleph_1)\\ &~\vee~ (\text{cardinality of each model} = \aleph_1 \wedge \text{cardinality of number of models} = \aleph_1). \end{align*}
Hence we have three "classes" of models that are each isomorphic within themselves.
But this looks terribly wrong. For example, is there a notion of multiplication within and between cardinalities? Could someone teach me how to think about different cardinalities of infinities?
Edit
With regard to the aleph-omega-1 case. My problem is I am not familiar with how to count up to this size. I see we have a continuum of options for the number of equivalent classes, and a continuum of options for sizes of each class so that the result sums up to $\aleph_{\omega_1}$. But if I would like to say something more satisfying than just "the number of equivalent classes is uncountable", how would I proceed?
Could someone to explain to me how to think about sets when they are this large. All I know about $\aleph_{\omega_1}$ is that is comes after this sequence:
$$\aleph_0,\aleph_1,\ldots,\aleph_{\omega},\aleph_{\omega+1},\ldots,\aleph_{\omega_1}.$$
Hardly enough knowledge to reason with.