On Coincidence of Dimensions in Closed Ordered Differential Fields

Abstract

Let $(R, \delta)$ be a closed ordered differential field, and $C$ its field of constants. In this note, we prove that for sets definable in the pair $(R, C)$, the $\delta$-dimension and the large dimension coincide. As an application, we characterize the definable sets that are internal to $C$, as those sets that are definable in $(R, C)$ and have $\delta$-dimension $0$. We further show that having $\delta$-dimension $0$ does not generally imply co-analyzability in $C$.