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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ \alpha_T$ is finite for $ \aleph_1$-categorical $ T$


Author: John T. Baldwin
Journal: Trans. Amer. Math. Soc. 181 (1973), 37-51
MSC: Primary 02H13; Secondary 02G20
MathSciNet review: 0319747
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1973-0319747-9
PII: S 0002-9947(1973)0319747-9
Keywords: $ {\aleph _1}$-categorical, strongly minimal, $ {\alpha _{{T^ \circ }}}$
Article copyright: © Copyright 1973 American Mathematical Society