TITLE: Definability in a given structure
QUESTION [8 upvotes]: I want to prove the following statements:

Is the function "sin" definable in the structure $(\Bbb{R},<,+,\cdot,0,1)$, that is does there exists a formula $\phi=\phi(x_0,x_1)$ such that for all $a,b\in\Bbb{R}$: $sin(a)=b$ iff $(\Bbb{R},<,+,\cdot,0,1)$ realizes $\phi[a,b]$ ?!
Is $\Bbb{Z}$ a definable subset of $(\Bbb{R},<,+,\cdot,0,1)$ ?
Is $\Bbb{Q}$ a definable subset of $(\Bbb{R},<,+,\cdot,0,1)$ ?

I have no idea how to solve this questions. I think i have to prove that i have to show that in a extension of the structure the subsets are not definable or i have to give a formula. Can someone help me?! Thanks

REPLY [3 votes]: An alternative approach is to start by proving that $M=(\Bbb{R},<,+,\cdot,0,1)$ eliminates quantifiers. It then easily follows that $M$ is o-minimal; that is, the only definable sets (even with parameters) are Boolean combinations of intervals. Thus $\Bbb{Q}$ and $\Bbb{Z}$ are not definable, and so neither is $\sin$, since a definition of $\sin$ would allow us to define $\Bbb{Z}$.