Arithmetical Definability over Finite Structures Troy Lee Abstract: Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform $\AC^0$ and $\FO(\PLUS,\TIMES)$. We prove finite analogs of three classic results in arithmetical definability, namely that $<$ and TIMES can first-order define PLUS, that $<$ and DIVIDES can first-order define TIMES, and that $<$ and COPRIME can first-order define TIMES. The first result sharpens the known equivalence ${\FO(\PLUS,\TIMES)=}{\FO(\BIT)}$ to $\FO(<,\TIMES)=\FO(\BIT)$, answering a question raised by Barrington et al. (LICS 2001) about the Crane Beach Conjecture. Together with previous results on the Crane Beach Conjecture, our results imply that $\FO(\PLUS)$ is strictly less expressive than $\FO(<,\TIMES)=\FO(<,\DIVIDES)=\FO(<,\COPRIME)$. In more colorful language, one could say this containment adds evidence to the belief that multiplication is harder than addition.