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 $\mathfrak{M}$?



Is "approximate categoricity" absolute?
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>


