How many language families are there? Let’s define a finite transducer as a $5$-tuple $(Q, A, B, \phi, \psi)$, where $Q$ is a finite collection of states, $A$ is a finite input alphabet, $\phi: Q\times A \to Q$ is the transition function and $\psi: Q \times A \to B^*$ is the output function.
Any transducer defines a transducer function $\overline{\psi}: Q\times A^* \to B^*$ described by the following recurrence:
$\overline{\psi}(q, \Lambda) = \Lambda$, where $\Lambda$ is the empty word.
$\overline{\psi}(q, a \alpha) = \psi(q, a) \overline{\psi}(\phi(q, a), \alpha)$, where $a \in A$, $\alpha \in A^*$. 
Let’s call a function $f: A^* \to B^*$ a regular transduction iff $\exists$ a finite transducer $(Q, A, B, \phi, \psi)$ and an initial state $q \in Q$, such that  $\forall \alpha \in A^*$ we have $f(\alpha) = \overline{\psi}(q, \alpha)$.
Now, let’s call a set of languages $\mathfrak{F}$ over a finite alphabet $A$ ($|A| > 2$) a family iff it satisfies two properties. 
1)$\forall L_1, L_2 \in \mathfrak{F} L_1 \cup L_2 \in \mathfrak{F}$
2)$\forall L \in \mathfrak{F}$ and $\forall$ regular transductions $f$ $f(L) \in \mathfrak{F}$.
My question is:

How many language families are there?

As $|A^*| = \aleph_0$, then there are $2^{\aleph_0}$ languages total, and thus the number of language families is $\leq 2^{2^{\aleph_0}}$.
On the other hand, it is $\geq 2^{\aleph_0}$ as every single language generates a countable family, and thus the union of all countable families (which is a proper subset of the set of all families) has size $2^{\aleph_0}$, which is only possible, when the number of countable language families is $2^{\aleph_0}$ itself.
However, I do not know how to determine, whether it is $2^{\aleph_0}$ or $2^{2^{\aleph_0}}$ (it can not be something in between because otherwise it would have been a constructive counterexample to the continuum hypothesis, which is known to be independent from ZFC).
 A: Note that transduction commutes with union, i.e., $$\tag1f(L_1\cup L_2)=f(L_1)\cup f(L_2).$$ It follows that the family $\langle \mathcal L\rangle$ generated by a set $\mathcal L$  of languages can be obtained by first taking all transduction results of all languages $L\in\mathcal L$, and only afterwards taking finite unions of these. Mixing the two constructions is not needed.
If we consider infinite words $\phi\in A^{\Bbb N}$ and a transduction $f$, it makes sense to speak of $f(\phi)$, which may be a finite or infinite word.
Let 
$$ \overline\phi=\{\,\psi\in A^{\Bbb N}\mid \exists f\colon f(\phi)=\psi\,\}$$
and 
$$ \widetilde\phi=\{\,\psi\in A^{\Bbb N}\mid \exists f\colon f(\psi)=\phi\,\}.$$
Clearly, $\overline\phi$ and $\widetilde\phi$ are  countable.
For $L\subseteq A^*$ and $\phi\in A^{\Bbb N}$, let $L(\phi)$ be the language consisting precisely of all prefixes $\phi(1)\phi(2)\ldots\phi(n)$ of $\phi$.
Write $L\rightsquigarrow\phi$ if $L\cap L(\phi)$ is infinite.
By Kőnig's lemma, each infinite language $L$ has at least one $\phi$ with $L\rightsquigarrow \phi$.
The following basic properties allow us to use  such $\phi$ to characterize language families:
One directly checks (a la Kőnig's lemma) that 
$$\tag2 f(L)\rightsquigarrow\psi\iff \exists \phi\colon \psi=f(\phi)\land L\rightsquigarrow \phi.$$
Also,
$$\tag3 L_1\cup L_2\rightsquigarrow\phi\iff  L_1\rightsquigarrow \phi\lor L_2\rightsquigarrow \phi.$$
With $\Omega$ standing for the first continuum-sized ordinal,
use transfinite recursion to pick a sequence $\{\phi_n\}_{n<\Omega}$ recursively such that $\phi_n\notin \bigcup_{k<n}\overline{\phi_k}\cup \bigcup_{k<n}\widetilde{\phi_k}$. This is always possible because there are only less than continuum-many $k<n$. 
By construction, 
$$\tag4\overline{\phi_n}\cap\overline{\phi_m}\ne\emptyset\iff n=m.$$
For each of the $2^{2^{\aleph_0}}$ subsets $S\subseteq \Omega$, consider the generated family
$$\mathfrak F_S:=\langle\{\,L(\phi_n)\mid n\in S\,\}\rangle.$$
By $(1)$, each $L\in\mathfrak F_S$ is of the form
$$L=f_1(\phi_{n_1})\cup\ldots\cup f_m(\phi_{n_m})$$
with $n_i\in S$.
Thus by $(2)$ and $(3)$,  $L\rightsquigarrow \phi$ implies $\phi\in \overline{\phi_{n_1}}\cup \ldots \cup\overline{\phi_{n_m}}$.
Using $(4)$, we conclude that
$$S=\{\,n\in \Omega\mid  \exists L\in\mathfrak F_S\colon L\rightsquigarrow \phi_n\,\} $$
can be reconstructed from $\mathfrak F_S$.
Hence there are at least $2^{2^{\aleph_0}}$ distinct language families.
