Exercise 2.5.5 david marker's [closed]

please help me to solve it,

Let $L = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $L$-theory of an equivalence relation with infinitely many infinite classes.

a) Write axioms for $T$.

b) How many models of $T$ are there of cardinality $\aleph_{0}$? $\aleph_{1}$? $\aleph_{2}$? $\aleph_{\omega_{1}}$?

c) Is $T$ complete?

I am eager to learn logic and this exercise!

The axioms are easy, for each $n$, the axioms

$$\exists x_1, \ldots , x_n(\wedge_{l\neq j} \neg x_i Ex_j)$$ $$\forall y\exists x_1, \ldots , x_n(\wedge x_i\neq x_j\wedge \wedge x_i Ey)$$

and of course the axioms saying $E$ is an equivalence relation.

In $\aleph_0$ there is only one model, with a countable number of classes, each one being countable. As a consequence the theory is complete.

For $\aleph_{\alpha}$, $0<\alpha$, let $\kappa_i$ be the number of classes with $\aleph_i$ many elements where $i\leq \alpha$. Then each $\kappa_1\leq |\alpha|+\aleph_0$ because there may be only a finite number of classes with a given cardinality. These numbers characterize the model. This if we look at models of size $\leq \aleph_{\alpha}$ there are $|\omega+\alpha|^{|1+\alpha|}$ many models. In particular for models of size $\aleph_0$ or $\aleph_1$ there are $\aleph_0^2=\aleph_0$ many models. For example a model could have $5$ countable classes and $\aleph_0$ many classes of size $\aleph_1$. Lastly remark that in this counting there are some that do not satisfy the axioms, eg a finite number of classes, but these will not affect the final result, as the true number of models will always be infinite.