How does categoricity interact with the underlying set theory?

Question

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?)

Answer 1

Categoricity is absolute.

By the Ryll-Nardzewski theorem, for a countable language, $\aleph_0$-categoricity of a complete theory $T$ is equivalent to $T$ proving for each natural number $n$ that there are only finitely many inequivalent formulas in $n$ variables. This property is evidently arithmetic and, thus, absolute.

Likewise, again in a countable language, it follows from the Baldwin-Lachlan theorem that a theory is categorical in some (hence, by Morley's theorem, all) uncountable cardinality just in case every model is prime and minimal over a strongly minimal set. Moreover, the strongly minimal formula may be taken to be defined over the prime model and the primality and minimality of every model over this strongly minimal formula is something which will be witnessed by an explicit analysis, hence, something arithmetic and absolute.

For uncountable languages, the situation is a little more complicated, but again categoricity is equivalent to an absolute property. Shelah shows that either the theory is totally transcendental and Morley's analysis in the case of countable languages applies, or the theory is strictly superstable though unidimensional.

Answer 2

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.

Answer 3

(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;

This might happen if you are interested in a (mildly) non-first order theory whose standard model of cardinality continuum, cf. the thesis of Martin Bays for an example of such an L\omega1\omega-theory. The question whether the theory is categorical in the cardinality of its standard model depends on the value of continuum, and it is much easier to prove when the continuum is \aleph_1....

Although I believe that some of the Scanlon's reply applies here as well, but you need some assumptions on your non-first order theory, e.g. assuming that the relevant AEC class is excellent.