$\alpha _T$ is finite for $\aleph _1$-categorical $T$
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- by John T. Baldwin PDF
- Trans. Amer. Math. Soc. 181 (1973), 37-51 Request permission
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 βOn strongly minimal setsβ by Baldwin and Lachlan. ${\alpha _T}$ is defined in βCategoricity in powerβ by Michael Morley.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 181 (1973), 37-51
- MSC: Primary 02H13; Secondary 02G20
- DOI: https://doi.org/10.1090/S0002-9947-1973-0319747-9
- MathSciNet review: 0319747