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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?

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up vote 5 down vote accepted

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.

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One then continues this in the presence of large cardinals through the projective hierarchy and beyond. –  Andres Caicedo Mar 3 at 1:11
    
@AndresCaicedo: Would you please explain more? –  user47697 Mar 10 at 13:53
    
Your answer is very nice, Prof. Marker. Would you please provide more examples (if there is any)? –  user47697 Mar 10 at 13:56
    
@CCD This is a standard result, and is not really model theoretic. Jech's set theory book should have at least a sketch and references. –  Andres Caicedo Mar 10 at 14:54

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