Suppose $\mathcal L \subseteq \mathcal L’$ are first-order languages, $\kappa$ is a cardinal, and $T’$ is a theory in $\mathcal L’$ that is $\kappa$-categorical. Let $T = T’ \restriction \mathcal L$. Is $T$ $\kappa$-categorical?

If $|\mathcal L’| = \kappa = \aleph_0$, then I can show the answer is yes using the Ryll-Nardzewski Theorem, which says that a countable theory is $\aleph_0$-categorical iff for each $n$, the number of $n$-types is finite.

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    $\begingroup$ I like this question quite a lot, since many of the categorical theories I considered at first do not seem directly to be counterexamples. $\endgroup$ – Joel David Hamkins May 22 at 9:16

The answer is no, one can lose categoricity in a reduct of a theory. Consider the following example.

Consider the theory $T$ describing a bijection between two disjoint infinite predicates $f:A\to B$. So a model consists of two disjoint parts, the $A$-part and the $B$-part, and a bijection $f$ between them. The language is $\{f,A,B\}$.

This theory is categorical in every cardinality. But if we restrict the theory to its consequences in the language with the two predicates $\{A,B\}$ and without the bijection, then it is just the theory of two disjoint infinite predicates, which is no longer categorical in uncountable powers, since one predicate could have a different cardinality than the other.

  • $\begingroup$ I'm confused. Take a model of the theory of size continuum with the two predicates countable and another where the two predicates have size continuum. You probably want to require that every object in the model satisfies one of the predicates as well. $\endgroup$ – Asaf Karagila Jun 1 at 21:23
  • $\begingroup$ Yes, that is what I had meant to convey. The model consists just of A, B and f. $\endgroup$ – Joel David Hamkins Jun 2 at 7:43

One can also break countable categoricity when $|\mathcal{L}'|>{\aleph_0}$. This example comes from an undergraduate course I took with Malliaris.

Let $\mathbb{P}$ be the collection of primes and let $\mathcal{P}(\mathbb{P})$ be the powerset of $\mathbb{P}$. Let $\mathcal{L'} = (+,\times,0,1;(D_{\alpha}(x))_{\alpha \in \mathcal{P}(\mathbb{P})})$ and let $T \models Th_{\mathcal{L}'}(\mathbb{N})$ with the usual interpretation of the symbols, and for each $\alpha \in \mathcal{P}(\mathbb{P})$, $\models D_{\alpha}(n)$ if and only if $n \in \alpha$. One can show that $T$ is $\aleph_0$-categorical and that the only countable model is the standard model. This follows from the fact that if one adds a single non-standard element, one must necessarily add $2^{\aleph_0}$ many elements.

However, if we let $\mathcal{L} = \{+\}$, then by Ryll-Nardzewski, we no longer have countable categoricity.


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