TITLE: Is "approximate categoricity" absolute?
QUESTION [11 upvotes]: Let $T$ be a countable first-order theory, and assume that $T$ has exactly one atomic model up to isomorphism in every uncountable cardinality. (By "atomic" I mean a model which omits the non-principal types).
Now let $\mathfrak{M}$ be a countable transitive model of set theory, and assume that $T$ is also (countable) in $\mathfrak{M}$. 
Is the above property preserved in $\mathfrak{M}$? i.e. does it hold in $\mathfrak{M}$ that there exists only one atomic model of $T$ in every uncountable cardinality in $\mathfrak{M}$?

REPLY [16 votes]: This is really a comment, but I need a bit more space.
If $\phi$ is a sentence of ${\cal L}_{\omega_1,\omega}$ that is $\aleph_0$-categorial, there is a complete first order theory $T$ in an expanded vocabulary such that the models of $\phi$ are exactly the reducts atomic models of $\phi$.  The expansion is done in such a way that two structures will be isomorphic in the original language if and only if they are isomorphic in the expanded language.
So your question is really the same as: For $\phi$ a sentence of ${\cal}L_{\omega_1,\omega}$ is ``$\phi$ is $\kappa$-categorical for all infinite
$\kappa$" absolute?
This is, as far as I know, still an open question.  It is also open if ``$\phi$ is $\aleph_1$-categorical" is absolute.
John Baldwin in his paper ``Amalgamation, Absoluteness, and Categoricity"
addresses some issues around this.  Here is a link
http://homepages.math.uic.edu/~jbaldwin/pub/singsep2010rev.pdf