Mathematics
Axiomatic Set Theory and the Zermelo–Fraenkel Axioms
Quick fact
The set of all sets that do not contain themselves cannot exist—this is Russell's Paradox, which led to the creation of axiomatic set theory in the early 20th century.
Why this is interesting
You've used sets since kindergarten—putting things in groups—but did you know that without a few strict rules, sets can create paradoxes that break mathematics itself?