Journal of Symbolic Logic

On Dedekind Complete O-Minimal Structures

Anand Pillay and Charles Steinhorn

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For a countable complete $o$-minimal theory $\mathbf{T}$, we introduce the notion of a sequentially complete model of $\mathbf{T}$. We show that a model $\mathscr{M}$ of $\mathbf{T}$ is sequentially complete if and only if $\mathscr{M} \prec \mathscr{N}$ for some Dedekind complete model $\mathscr{N}$. We also prove that if $\mathbf{T}$ has a Dedekind complete model of power greater than $2^{\aleph_0}$, then $\mathbf{T}$ has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory--namely, a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables--that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers.

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J. Symbolic Logic, Volume 52, Issue 1 (1987), 156-164.

First available in Project Euclid: 6 July 2007

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Pillay, Anand; Steinhorn, Charles. On Dedekind Complete O-Minimal Structures. J. Symbolic Logic 52 (1987), no. 1, 156--164.

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