Journal of Symbolic Logic

On Ehrenfeucht-Fraisse Equivalence of Linear Orderings

Juha Oikkonen

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C. Karp has shown that if $\alpha$ is an ordinal with $\omega^\alpha = \alpha$ and $A$ is a linear ordering with a smallest element, then $\alpha$ and $\alpha \bigotimes A$ are equivalent in $L_{\infty\omega}$ up to quantifer rank $\alpha$. This result can be expressed in terms of Ehrenfeucht-Fraisse games where player $\forall$ has to make additional moves by choosing elements of a descending sequence in $\alpha$. Our aim in this paper is to prove a similar result for Ehrenfeucht-Fraisse games of length $\omega_1$. One implication of such a result will be that a certain infinite quantifier language cannot say that a linear ordering has no descending $\omega_1$-sequences (when the alphabet contains only one binary relation symbol). Connected work is done by Hyttinen and Oikkonen in [H] and [O].

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J. Symbolic Logic, Volume 55, Issue 1 (1990), 65-73.

First available in Project Euclid: 6 July 2007

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Oikkonen, Juha. On Ehrenfeucht-Fraisse Equivalence of Linear Orderings. J. Symbolic Logic 55 (1990), no. 1, 65--73.

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