Mathematics Asked by user826451 on January 12, 2021

Morley’s theorem says if $T$ is a complete theory in a countable language and $T$ is $kappa$-categorical for some uncountable $kappa$, then $T$ is $kappa$-categorical for any uncountable $kappa$. Shelah generalized this theorem to uncountable languages, but I can’t find the exact statement.

In particular, if $T$ is $kappa$-categorical for some $kappa$ greater (or also equal?) than $|T|$, then $T$ is $kappa$-categorical for every $kappa$ greater (or also equal) than $|T|$?

The wikipedia page for Categorical theory answers your question (emphasis mine):

Saharon Shelah extended Morley's theorem to uncountable languages: if the language has cardinality $kappa$ and a theory is categorical in some uncountable cardinal

greater than or equal to$kappa$ then it is categorical in all cardinalitiesgreater than$kappa$.

Now this is actually a bit stronger than you might expect! Morley's theorem says that if a theory in a countable language is categorical in a cardinal **greater than** $aleph_0$, then it is categorical in all cardinalities **greater than** $aleph_0$. On the other hand, if a theory in a countable language is categorical in a cardinal **equal to** $aleph_0$ (i.e. the theory is countably categorical), this does not guarantee categoricity in any other cardinals.

Shelah's possibly surprising result, as stated above, puts together two theorems from Chapter IX of *Classification Theory* (pages 490 and 491). Theorem 1.16 is the natural generalization of Morley's theorem, and Theorem 1.19 deals separately with the case of a theory $T$ which is $|T|$-categorical, showing that this case trivializes when $|T|$ is uncountable.

THEOREM 1.16: Suppose $T$ is categorical in some $lambda > |T|$ or every model of $T$ of cardinality $lambda$ (for some $lambda>|T|$) is $|T|^+$-universal. Then $T$ is categorical in every $mu > |T|$, and every model of $T$ of cardinality $>|T|$ is saturated.

THEOREM 1.19: If $T$ is categorical in $|T|>aleph_0$, then $T$ is a definitional extension of some $T'subseteq T$, $|T'|<|T|$.

The point is that if $T$ is a definitional expansion of $T'$, then there is a one-to-one cardinality-preserving correspondence between the models of $T$ and the models of $T'$. If $T$ is $|T|$-categorical, then $T'$ is also $|T|$-categorical. Since $|T'|<|T|$, by Theorem 1.16, $T'$ is $kappa$-categorical for all $kappa > |T'|$. So also $T$ is $kappa$-categorical for all $kappa>|T'|$, and in particular for all $kappa>|T|$.

What's happened here is that the uncountable $|T|$-categorical theory $T$ is "secretly" just a theory of smaller cardinality. And this is really quite a silly situation to be in! To say that $T$ is a definitional expansion of $T'$ is to say that every symbol in the language of $T$ which is not in the language of $T'$ is defined by a formula in the language of $T'$. But there are only $|T'|$-many formulas in the language of $T'$, so while there are $|T|$-many new symbols, up to equivalence there are only $|T'|$-many!

To give an explicit example, we could take $T'$ to be the theory of algebraically closed fields and take $T$ to be the theory obtained by introducing uncountably many constant symbols ${c_alphamid alphain kappa}$ and setting them all to $0$ by adding axioms $c_alpha = 0$ for all $alpha$. Theorem 1.19 says that every example has to be almost as trivial as this one. Upshot: the behavior of countable $aleph_0$-categorical theories is very special to the countable.

Answered by Alex Kruckman on January 12, 2021

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