Anyone who has tried to teach Calculus I students the definition of the limit knows that students have a difficult time working out definitions with multiple quantifiers. In fact it is something that we must come back to again and again with our mathematics majors.
While doing a little research this week I came across the definition of something called the specification property for a discrete dynamical system—it was introduced in the early 1970’s by Rufus Bowen. The logical complexity of the definition made me laugh out loud the first time I read it. I had to reread it about 5 times to figure out what it was saying.
The dynamical system is said to satisfy the specification property if for any there exists an integer such that for any , any points , any integers with for and for any integer with , there exists a point with such that for , .
(In case you’re wondering, roughly speaking it says that for a dynamical system with this property, given a finite number of finite orbit segments, there is a periodic orbit that follows one orbit segment very closely, then a little while later follows the next one, then a little while later follows the next one, etc.)