TITLE: How does categoricity interact with the underlying set theory?
QUESTION [11 upvotes]: Here's the setup: you have a first-order theory T, in a countable language L for simplicity. Let k be a cardinal and suppose T is k-categorical. This means that, for any two models
M,N |= T
of cardinality k, there is an isomorphism f : M --> N.
Supposing all this happens inside of ZFC, let's say I change the underlying model of ZFC, e.g by restricting to the constructible sets, or by forcing new sets in. I would like to understand what happens to the k-categoricity of T.
I'll assume the set theory doesn't change so drastically that we lose L or T. Then, a priori, a bunch of things may happen:
(i) We may lose all isomorphisms between a pair of models M,N of cardinality k;
(ii) Some models that used to be of cardinality k may no longer have bijections with k;
(iii) k may become a different cardinal, meaning new cardinals may appear below it, or others may disappear by the introduction of new bijections;
(iv) some models M, or k itself, may disappear as sets, leading to a new set being seen as "the new k".
Overall, nearly every aspect of the phrase "T is k-categorical" may be affected. How likely is it to still be true? Do some among (i)-(iv) not matter, or is there some cancellation of effects? (Say, maybe all isomorphisms M-->N disappear, but so do all bijections between N and k?)

REPLY [9 votes]: In contrast to absoluteness of categoricity in First Order Logic, there are many interesting non-absoluteness phenomena at low infinite cardinals outside of first order. Perhaps the most important is the following:
(Under wGCH at $\kappa$, i.e. $2^\kappa<2^{\kappa^+}$) For every abstract elementary class $(\mathfrak K,\prec_{\mathfrak K})$ with $LS(\mathfrak K)\leq \kappa$, if $\mathfrak K$ is categorical in $\kappa$ and fails to have the amalgamation property at $\kappa$ ($AP_\kappa$), then $\mathfrak K$ is not categorical in $\kappa^+$ (indeed, it has the maximum number of models of size $\kappa^+$, $2^{\kappa^+}$).
In contrast, Martin's Axiom provides a completely different picture:
There exists an AEC (axiomatizable in $L_{\omega_1,\omega}(Q)$) $\mathfrak K_r$ with $LS(\mathfrak K_r)=\aleph_0$, $\mathfrak K_r$ is categorical in $\aleph_0$ and fails $AP_{\aleph_0}$ that is categorical in $\aleph_1$ in the presence of $MA_{\aleph_1}$.
The theorem and the example are due to Shelah (but have had improved presentations due to various other authors - Grossberg and Baldwin most prominent). Notice that here categoricity of a certain class in $\aleph_1$ is NOT absolute. Many open questions remain.