Vaught's Conjecture is a dual form of Continuum Hypothesis in model theory. It asserts that for each complete consistent theory $T$ in a countable language if $I(T,\aleph\_{0})>\aleph\_{0}$ then $I(T,\aleph\_0)=2^{\aleph\_{0}}$.


The best known result here is proved by Morley.


**For each complete theory $T$ in a countable language if $I(T,\aleph\_{0})>\aleph\_{1}$ then $I(T,\aleph\_0)=2^{\aleph\_{0}}$.**


We call this kind of overspill from $\aleph\_1$ to $2^{\aleph\_0}$, a Morley phenomena.


**Question.** For which non-trivial properties $P$ is the following true?


**If $\mathcal{F}$ is a family of reals (subsets of $\omega$) with $|\mathcal{F}|>\aleph\_{1}$ and each set $X\in \mathcal{F}$ satisfies the property $P$ then $|\mathcal{F}|=2^{\aleph\_0}$.**


In the other words, for which kind of reals do we have a Morley phenomena for families of that particular type of reals? Is there any correspondence between these type of reals and countable models of a complete theory in a countable language? 



Morley Phenomena for Special Families of Reals
The classical example is ${\mathbf\Pi}^1\_1$ sets of reals.
We say that $X\subset \mathbb R$ is ${\mathbf\Pi}^1\_1$ if the complement of the projection
of a Borel set in ${\mathbb R}^n$. Any such set of cardinality greater than $\aleph\_1$ has cardinality continuum.


