Journal of Symbolic Logic

On Scott and Karp Trees of Uncountable Models

Tapani Hyttinen and Jouko Vaananen

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Let $\mathfrak{U}$ and $\mathfrak{B}$ be two countable relational models of the same first order language. If the models are nonisomorphic, there is a unique countable ordinal $\alpha$ with the property that $\mathfrak{U} \equiv^\alpha_{\infty\omega} \mathfrak{B} \text{but not} \mathfrak{U} \equiv^{\alpha + 1}_{\infty\omega} \mathfrak{B},$ i.e. $\mathfrak{U}$ and $\mathfrak{B}$ are $L_{\infty\omega}$-equivalent up to quantifier-rank $\alpha$ but not up to $\alpha + 1$. In this paper we consider models $\mathfrak{U}$ and $\mathfrak{B}$ of cardinality $\omega_1$ and construct trees which have a similar relation to $\mathfrak{U}$ and $\mathfrak{B}$ as $\alpha$ above. For this purpose we introduce a new ordering $T \ll T'$ of trees, which may have some independent interest of its own. It turns out that the above ordinal $\alpha$ has two qualities which coincide in countable models but will differ in uncountable models. Respectively, two kinds of trees emerge from $\alpha$. We call them Scott trees and Karp trees, respectively. The definition and existence of these trees is based on an examination of the Ehrenfeucht game of length $\omega_1$ between $\mathfrak{U}$ and $\mathfrak{B}$. We construct two models of power $\omega_1$ with $2^{\omega_1}$ mutually noncomparable Scott trees.

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J. Symbolic Logic, Volume 55, Issue 3 (1990), 897-908.

First available in Project Euclid: 6 July 2007

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Hyttinen, Tapani; Vaananen, Jouko. On Scott and Karp Trees of Uncountable Models. J. Symbolic Logic 55 (1990), no. 3, 897--908.

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