Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture. 


First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a countable language has infinitely many models of size $\aleph\_0$, then either $I(T,\aleph\_0)=\aleph\_0$ or $I(T,\aleph\_0)=2^{\aleph\_0}$. We will denote this conjecture as $VC$. 


It has been shown that if $I(T,\aleph\_0)> \aleph\_1$, then $I(T,\aleph\_0)=2^{\aleph\_0}$. Therefore, we know that $ZFC+CH \vdash VC$. 


If one wants to prove that $VC$ is false, then one must construct a model $M$ such that $M\models ZFC$ and a theory $T\_1$, such that $M\models ZFC+ \neg CH$ and $M \models I(T\_1,\aleph\_0)=\aleph\_1$. 


But here lies my problem: Let $M\_1,M\_2 \models ZFC + \neg CH$. Suppose that we can construct first order theories $T\_1$ and $T\_2$ such that $M\_1 \models I(T\_1,\aleph\_0)=\aleph\_1$ and $M\_2 \models I(T\_2,\aleph\_0)=\aleph\_1$. Is it possible that $M\_1\models I(T
\_2,\aleph\_0)=2^{\aleph\_0}$ and $M\_2\models I(T\_2,\aleph\_0)=2^{\aleph\_0}$? 


Better yet, we can simply ask: if $T$ is a counterexample to Vaught's Conjecture in some model $M$ of $ZFC$ must it be the case that $N\models I(T,\aleph\_0)=\aleph\_1$ for any $N$ such that $N\models ZFC$?


Thanks!



How do we know if Vaught's Conjecture is Absolute?
The answer is yes. See section 5 of the paper "Bounds on weak scattering" by Sacks (<http://www.math.harvard.edu/~sacks/bws.pdf>); he cites Morley 1970 ("The number of countable models," <http://www.jstor.org/stable/2271150>) as the original proof.


