Model theory/stability theory

Question

I was flipping through Baldwin's Stability Theory book and saw an example that has me confused...

The example is a 1st-order theory $T$ of refining equivalence relations $E_i(x, y), i< ω$, where

• $E_j$ refines $E_i$ when $i < j$
• each $E_{i+1}$ relation splits each $E_i$-equivalence class into two $E_{i+1}$-equivalence classes.

Here's what I'm picturing: $E_0$ splits the model into two classes, call them $a_0/E_0$ and $a_1/E_0$. Then $E_1$ splits $a_0/E_0$ into two pieces $a_{00}/E_1$ and $a_{01}/E_1$, and $E_1$ splits the class $a_1/E_0$ into two pieces $a_{10}/E_1$ and $a_{11}/E_1$, ad infinitum.

So the classes are arranged like the tree ${}^{<ω}2$ of finitely long $0-1$ sequences (the tree ordering is sequence extension), where the $n$th level of the tree has the $E_n$ classes as nodes.

My confusion:

1. Each infinite branch of the tree (there are $2^\omega$ of them) should correspond to an incomplete type of an element, right? And those types are pairwise contradictory. So, if $M$ is a countable model of $T$, there are $2^ω$ types over $M$. If all that is right, then $T$ is $\omega$-unstable. $T$ is supposed to be (some kind of) stable however.
2. But can't the type space still blow up over uncountable sets of parameters for cardinals of countable cofinality? Is it that such a T is stable at some cardinals, but doesn't “stay stable”?

Sorry if this is boring or obtuse! -M

Answer 1

This theory is stable in all cardinals that are at least $\mathfrak c$. To see this, notice that a type is determined entirely by which class of each $E_n$ it belongs in (of which there are $\mathfrak c$ many possible combinations in total) and which elements it is (not) equal to.

Such a theory is called superstable.

There are theories which are stable, but not superstable, and do explode precisely for parameters of size $\kappa$ such that $\kappa^\omega>\kappa$. For example, the theory of (countably) infinitely many independent equivalence relations with infinitely many classes. With $\kappa$ parameters you can have each relation have $\kappa$ many classes, yielding $\kappa^\omega$ types.

(Edit: the theory I mentioned is of infinitely many independent relations with infinitely many classes; the theory of infinitely many independent equivalence relations with two classes is, I believe, bi-interpretable with the one you described in your question.)