Show that if $L$ is countable and contains a two-place predicate symbol, there are $2^{2^{\aleph_0}}$ classes of $L$-structures closed under $\equiv$ We say that a class of structures $K$ is closed under elementary equivalence ($\equiv$) if for all $A, B$, if $A \in K$ and $A \equiv B$, then $B \in K$.  How to show that if $L$ (as a set of specific symbols) is countable and contains a two-place predicate symbol, then there are $2^{2^{\aleph_0}}$ classes of structures for $L$ closed under $\equiv$ but only $2^{\aleph_0}$ such $EC_\Delta$ classes?
Here is what I think. Every $EC_\Delta$ class is obtained via a set of first-order sentences. If $L$ is countable, there are $2^{\aleph_0}$ such sets because $card(Sent(L)) = \aleph_0$ and $\wp(Sent(L)) = 2^{card(Sent(L))}$. And every $EC_\Delta$ class is closed under $\equiv$.
But how to show that there are $2^{2^{\aleph_0}}$ classes of $L$-structures closed under $\equiv$? Why the assumption that $L$ contains a two-place predicate symbol is relevant?
Thanks for help!
 A: I will write "many" for the cardinal of the power set of the reals.
Note that a class of structures which is closed with respect to elementary equivalence is the same as a collection of complete theories (exactly those theories which are the theory of some structure in your collection).
If you want many classes, it is enough to have continuum many complete theories: for each subset of these theories, look at the class of those structures $M$ with the property that the theory of $M$ is in the subset. Different subsets give different collections. So let us show how to construct continuum many complete theories (and explain why you need to assume something about the language)
If $L$ has a binary predicate $E$, you can say that $E$ is an equivalence relation, and then for every infinite subset $A$ of the natural numbers you can say that there is precisely one equivalence classes of size $k$, for each $k$ in $A$. This gives you continuum many distinct theories, since there are continuum many infinite susets of the natural numbers.
If $L$ has a single unary predicate $P$ then you cannot say much. You can only specify the number of elements satisfying $P$ (finite for some fixed natural number, or infinite) and similarly for the complement. So there are only countably many complete theories in this language.
EC classes are those classes of structures which satisfy some given $L$-theory, not necessarily complete. These are in bijective correspondence with $L$-theories, again not necessarily complete, as before. Since an $L$-theory is just a collection of sentences and there are only countably many sentences, there can only be continuum many theories, and hence only continuum many EC classes. Conversely, there are continuum many complete theories, and the class of models of each of these complete theory is an EC class. These classes are all distinct, so there are continuum many classes.
