This is very much the same post as I posted at math.stackexchange.

I am following the definitions of an o-minimal system in "Tame Topology and O-minimal Structures" by Lou van den Dries.

It is immediate from the definition that the graph of $\sin(x)$ is not a tame set (intersect it with $y=0$). But what about a slightly rotated one? Or one which is both rotated and translated. To me they look to be tame (unless rotated by $\pi/4$). Is it correct that these sets are contained in *some* o-minimal system? And how can I easily 'recognize' tame sets? E.g. my intuition is that a collection of sets in $\mathbb{R}^2$ are tame if they do not invalidate the minimality axiom ($S_1$ contains exactly finite unions of points and open intervals). If so I can just complete with whatever sets needed in order for it to be a o-minimal structure.

I am familiar with the monotonicity theorem and how that may obstruct a cell decomposition of the aforementioned sets but I would really like to see from the very definition what goes wrong. It would be no problem if all lines in $\mathbb{R}^2$ were included but I cannot see how that is the case (all horizontal, vertical and $y=x$ are included by definition).