TITLE: First order definition of $\pi$
QUESTION [19 upvotes]: $e$ has a very short first-order definition. It's the only constant that makes this true:
$$\forall x,e^x\ge x+1$$
What about $\pi$? What's the shortest first-order definition we could give it?
I'm including things like addition, subtraction, multiplication, division, (real) exponentiation, roots, inequalities, etc. $\forall$ and $\exists$ range over $\Bbb R$. If you want to use differentiation or limits, expand it into its $\epsilon-\delta$ form.

REPLY [2 votes]: $ \pi < 4$ and for every integer $n \ge 3$ there is a complex $n$-th root of unity $\omega_n$ such that $$n |\omega_n -1| < 2\pi < (n+1) |\omega_n -1|$$