## Peter’s Solution to Exercise 1

Assume $\mathcal M$ and $S$ satisfy all of the assumptions in the exercise, including 1-4. For each $m\in\mathbb N$, define $A_m = \{x\in\Pi_n(S) : |S_x| \geq m\}$. We will show that each $A_m$ is either empty or equal to $S$, and hence that $S_x$ is equal to the minimal value of $m$ such that $A_m\neq\emptyset$.…

## O-minimal structures

Let ${\cal M}$ be an expansion of a dense linear order $(M,\lt)$. We call ${\cal M}$ o-minimal if every definable subset of $M$ is a finite union of points and intervals. Examples (without details) By quantifier elimination, every dense linear order without endpoints is o-minimal. Let ${\cal V} = (V,\lt,+,(\lambda_k)_{k \in K})$ be an ordered…