Reconciling two different o-minimality definitions In Tame topology and o-minimal structures by L. van den Dries there's the definition of an o-minimal structure using a sequence (S_n) that satisfies certain properties, and in model theory books there's the definition of an o-minimal structure using definable sets and formulas.
(https://en.wikipedia.org/wiki/O-minimal_theory) presents these two definitions together.
Can someone please explain why these two definitions are equivalent?
I can understand the definable sets of a model with an o-minimal theory form an o-minimal structure, in the sense of van den Dries, but I can't figure out how an o-minimal structure (S_n) gives rise to an o-minimal model!
 A: Let $(M,(S_n)_{n\in \omega})$ be an o-minimal structure. 
Let $L$ be the language containing an $n$-ary relation symbol $R_A$ for every set $A\in S_n$, and interpret these symbols in $M$ in the obvious way, i.e. for $A\in S_n$, the symbol $R_A$ is interpreted as $A\subseteq M^n$.
Now you can prove by induction that every definable set $X\subseteq M^n$ is in $S_n$. Definable sets are defined by formulas, which are built up from atomic formulas by Boolean combinations and quantification.
In the base case, the sets defined by the atomic formulas $R_A(x_1,\dots,x_n)$ and $x_1 = x_2$ are in $S_n$ by definition (for equality, we use closure property 3). Of course, we can plug in other tuples of variables (i.e. $R_A(x_2,x_1)$ instead of $R_A(x_1,x_2)$), and we can think of atomic formulas as living in larger variable contexts. This closure under dummy variables and variable substitution is handled by closure property 2 and the Lemma 2.2 in van den Dries' book (using a trick involving projection).
Then closure under Boolean combinations is property 1, and quantification is handled by closure property 4.
Once you see that the definable sets in this structure are exactly the sets in the families $S_n$, it's easy to see that Wikipedia's conditions 5 and 6 correspond exactly to o-minimality of the model theoretic structure $\langle M,(R_A)_{A\in \bigcup_\omega S_n}\rangle$
