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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?)

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3 Answers 3

up vote 19 down vote accepted

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.

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1  
Just to clarify a bit further: what's going on here is that you can use Schoenfield's absoluteness theorem: Pi^1_2 statements are absolute between transitive models of ZFC which contain all the same ordinals (so this takes care of forcing or restricting to L). So the claim is that all the properties mentioned in this answer can be written in the form: "For any subset X of $\omega$, there is a subset $Y$ of omega such that ..." followed by a statement where all quantifiers range over either $\omega$ or the theory $T$ you're talking over. –  John Goodrick Apr 7 '10 at 6:30
    
...and I do believe Tom's answer is right (and it's a good answer), but checking the absoluteness of properties like "T is totally transcendental" always gives me a little pause. Good exercises for a set theory class! –  John Goodrick Apr 7 '10 at 6:32
    
Hmm, actually the situation for uncountable languages isn't clear to me. Unlike in the case where T is countable, you can no longer describe categoricity by quantifying over countable models (since there might not be any!) and so I don't see how to apply Shoenfield absoluteness. –  John Goodrick Apr 7 '10 at 6:54
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@John: The key is that Morley rank (and all good ordinal rankings) is absolute for models that have the same ordinals. This is also the key idea behind Shoenfield absoluteness. IIRC, another way to see it is that scattered compact Hausdorff spaces are also absolute. –  François G. Dorais Apr 7 '10 at 13:42
    
Francois: thanks for the clarification about totally transcendence. I'm satisfied now that this is absolute, but still curious about what Tom had in mind for the case where L is uncountable (is there a stronger absoluteness theorem we could use?). –  John Goodrick Apr 7 '10 at 14:29

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.

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Hola Andrés. Bienvenido! –  Andres Caicedo Feb 29 '12 at 19:02
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Welcome, Andrés, inventor of unfoldable and strongly unfoldable cardinals! –  Joel David Hamkins Feb 29 '12 at 20:23
    
Hi Andres! Great to see you here! –  Todd Eisworth Mar 1 '12 at 0:53

(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.

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Hi mmm, thanks for your answer. I have only vague familiarity with non-first-order systems, but a theory whose categoricity depends on the value of c would certainly shed light on the tamer case of FOL. –  Pietro KC Jun 3 '10 at 6:13

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