An $\omega$-stable theory that is not uncountably categorical

equivalence-relationslogicmodel-theory

Morley's theorem states that a countable complete theory that's $\omega$-stable and has no Vaughtian pair is uncountably categorical. From the statement, I guess the "no Vaughtian pair" condition is necessary. In searching for examples of theories that are $\omega$-stable (with Vaughtian pair) but not uncountably categorical, I stumbled upon this article. An example is given in Example 3.1.3, namely the theory in the language $\{E\}$ saying that $E$ is an equivalence relation and that for each natural number $n$, there is one $E$-class of size $n$. However, I don't understand the argument in the article, so I might just ask for an explanation here.

Why is this theory $\omega$-stable but not uncountably categorical?

Best Answer

Call this theory "$T$."

Showing that $T$ is not uncountably categorical is straightforward. Keep in mind that a model of $T$ is gotten by adjoining to the prime model (= one equivalence class of size $n$ for each finite $n$) an arbitrary number (possibly $0$) of equivalence classes of arbitrary infinite cardinalities. Now given $\kappa$ infinite, the following are some examples of non-isomorphic models of $T$ with cardinality $\kappa$:

  • Exactly one class of size $\kappa$, no other infinite classes.

  • Exactly two classes of size $\kappa$, no other infinite classes.

  • For $\kappa>\aleph_0$: exactly one class of size $\kappa$, exactly one class of size $\aleph_0$, no other infinite classes.

And so on. Personally I think the last example is especially important; it's good to keep in mind that similar-looking "pieces" of the model (e.g. infinite classes) can behave very differently.

As for $\omega$-stability, let's start by considering the no-parameters case: can we show that there are only countably many $1$-types over $\emptyset$?

(As usual we're working in some "sufficiently saturated" monster model $\mathfrak{M}$ here.)

Well, intuitively the $1$-type over $\emptyset$ of some element $a$ is determined by a simple cardinal invariant $size(a)$:

Take $size(a)$ to be the size of the equivalence class $a$ lives in.

After proving that in fact $size(a)=size(b)$ implies $tp_\emptyset(a)=tp_\emptyset(b)$, we can now count the number of possible values of this invariant within the monster model $\mathfrak{M}$:

Either $size(a)$ is finite, or by saturation $size(a)=\vert\mathfrak{M}\vert$. So there are $\aleph_0$-many possible values of $size(a)$.

We now need to think about folding in parameter sets larger than $\emptyset$ and tuple lengths larger than $1$ - that is, counting $n$-types over $A$ for arbitrary finite $n$ and countable $A\subseteq\mathfrak{M}$. EDIT: not really, see Alex Kruckman's comment below; I'm making things harder than they need to be here. This gets a bit tedious, but the basic idea is still the same: we want to show that there are only a few different possible behaviors for a given tuple.

For example, here's how to count the $2$-types over a one-element parameter set $\{c\}$. The type of a pair $(a_0,a_1)$ in $\mathfrak{M}$ over $\{c\}$ is determined by the following data.

First, we have the "$\emptyset$-part" of each individual element of the pair $(a_0,a_1)$:

What are the cardinalities $\alpha_0$, $\alpha_1$ of the classes of $a_0,a_1$ in $\mathfrak{M}$? As before, there are $\aleph_0$-many possibilities here.

Next, we consider the structure within the tuple $(a_0,a_1)$:

Are $a_0$ and $a_1$ equal? Are $a_0$ and $a_1$ equivalent? There are finitely many possibilities here.

Finally, we bring the parameter into the question:

How do $a_0$ and $a_1$ compare with $c$, both with respect to equality and equivalence? Again, there are only finitely many possible behaviors.

Putting all this together we get as hoped for only countably many $2$-types over a one-element parameter set in $\mathfrak{M}$. And it should be reasonably clear how to continue this to get $\omega$-stability.

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