$\alpha _T$ is finite for $\aleph _1$-categorical $T$

Abstract:

Let $T$ be a complete countable ${\aleph _1}$-categorical theory. Definition. If $\mathcal {A}$ is a model of $T$ and $A$ is a $1$-ary formula in $L(\mathcal {A})$ then $A$ has rank 0 if $A(\mathcal {A})$ is finite. $A(\mathcal {A})$ has rank $n$ degree $m$ iff for every set of $m + 1$ formulas ${B_1}, \cdots ,{B_{m + 1}} \in {S_1}(L(\mathcal {A}))$ which partition $A(\mathcal {A})$ some ${B_i}(\mathcal {A})$ has rank $\leqslant n - 1$. Theorem. If $T$ is ${\aleph _1}$-categorical then for every $\mathcal {A}$ a model of $T$ and every $A \in {S_1}(L(\mathcal {A})),A(\mathcal {A})$ has finite rank. Corollary. ${\alpha _T}$ is finite. The methods derive from Lemmas 9 and 11 in “On strongly minimal sets” by Baldwin and Lachlan. ${\alpha _T}$ is defined in “Categoricity in power” by Michael Morley.