# Is “approximate categoricity” absolute?

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 $\mathfrak{M}$?

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I may be missing something obvious, but could the close voters explain what they think is wrong with the question? – Emil Jeřábek Dec 28 '13 at 20:16
Related question: mathoverflow.net/questions/20590 . – Emil Jeřábek Dec 28 '13 at 20:31

## 1 Answer

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

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I think this is absolutely valuable as an answer, not a mere comment. – Todd Trimble Dec 30 '13 at 19:00