Definable closure

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. For $A \subseteq M$, the definable closure of $A$ is defined by $$\dcl(A):= \set{b \in M:\ \{b\} \text{ is } A\text{-definable}}.$$ Exercise Let $A \subseteq M$. Prove that $\dcl(A) = \acl(A)$. Let $\phi(x)$ be a formula with parameters in $A$. Prove that…

An “ordered Ramsey” theorem

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$, and let $I$ be an open interval. Ordered Ramsey Theorem (Peterzil and Starchenko) Let $S_1, \dots, S_k \subseteq M^2$ be definable, and assume that $I^2 \subseteq S_1 \cup \cdots \cup S_k$. Then there exist $l \in \{1, \dots, k\}$ and an open…

Reduction of the Monotonicity Lemma

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. Let $f:I \longrightarrow M$ be definable, with $I = (a,b)$ an interval in $M$. We start by translating the condition “$f$ is strictly increasing” into a condition involving two subsets of $I^2$: on the one hand, we have the triangle $\displaystyle \Delta(I):=…

The Monotonicity Theorem

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. Let $f:I \longrightarrow M$ be definable, with $I = (a,b)$ an interval in $M$. Definition We call $f$ strictly monotone if $f$ is either constant, or strictly increasing, or strictly decreasing. Our first goal is to prove the Monotonicity Theorem There are…

O-minimality and uniform finiteness

Let ${\cal M}$ be an o-minimal expansion of a dense linear order $(M,\lt)$. The first big question about o-minimality is the following: is o-minimality an elementary property, that is, given ${\cal N} \equiv {\cal M}$, is ${\cal N}$ necessarily o-minimal? Lemma The following are equivalent: every ${\cal N} \equiv {\cal M}$ is o-minimal; for every…

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