Let $L$ be the language $\{R\}$ containing a single binary relation symbol. Consider the space $S_0(L)$ of complete, first-order $L$-theories. This is a seperable, compact Hausdorff space; what is its Cantor-Bendixson rank?
More precisely, define $X_\alpha \subseteq S_0(L)$ via: $X_0 = S_0(L)$, $X_{\alpha+1} = X_\alpha \backslash$ the isolated points of $X_\alpha$, and take intersections at limit stages. There will be some least $\alpha_0 < \omega_1$ with $X_{\alpha_0}$ perfect. So the question is:
What is this $\alpha_0$?
Some observations:
- A theory $T \in S_0(L)$ is Cantor-Bendixson rank $0$ iff it is finitely axiomatizable; it is rank $1$ iff it is finitely axiomatizable over the axiom schema "I am not finitely axiomatizable," etc.
- If we had chosen an infinite language then the Cantor-Bendixson rank would be $0$. Any finite language $L'$ containing a binary relation symbol has the same Cantor-Bendixson rank as $S_0(L)$, since $S_0(L)$ and $S_0(L')$ can be embedded as clopen subsets of each other.
The following is all I know so far:
Fact. $\alpha_0 \geq \omega$.
Proof. (Rewritten in more detail.) Fix $n$, and let $\phi$ be the sentence "$R$ is an equivalence relation with $n$ classes." Then we can write down all the complete extensions of $\phi$; namely let $(m_0, m_1, \ldots, m_{l-1})$ be a sequence of numbers of length $l \leq n$, and let $T_{\overline{m}}$ be the theory of the model $M_{\overline{m}}$ with $n$ equivalence classes $(A_0, \ldots, A_{n-1})$, with $A_i$ having size $m_i$ for $i < l$, and $|A_i| = \infty$ for $i \geq l$.
I claim that the Cantor-Bendixson rank of $T_{\overline{m}}$ is $k := n-|\overline{m}|$, i.e. $k = $ the number of infinite classes.
If $k = 0$ then $T_{\overline{m}}$ is finitely axiomatizable clearly. Conversely, an easy compactness argument shows that whenever $k > 0$, $T_{\overline{m}}$ cannot be finitely axiomatizable. (Same proof as that the theory of infinite sets is not finitely axiomatizable.)
Suppose we have verified the claim for $k < n$, and ${\overline{m}}$ has length $n-k-1$. Then consider the sentence $\psi := $"$R$ is an equivalence relation, it has $n$ classes, and there exist $a_0, \ldots, a_{n-k-1}$ all $R$-inequivalent, so that $a_i/R$ has size $m_i$." Then this axiomatizes $T_{\overline{m}}$ over the negation of all theories of rank $\leq k$: for suppose $M \models \psi$, with rank(Th($M)) \geq k+1.$ Then $M$ must have at least $k+1$ infinite classes, which together with $\psi$ yields that $M \models T_{\overline{m}}$.
And a similar compactness argument gets that these are all of the rank $k+1$ theories.
