In order to prove any hypothesis false, it is enough to find a single counterexample. Assuming such counter-examples exist and are discrete, we can conclude from the well-ordering axiom that one of them must be the smallest. This smallest counter-example is sometimes called the minimal criminal. Furthermore, in order to prove our hypothesis true, it is enough to show that there is no such minimal criminal. After all, if no counter-example is smallest, there can be no counter-example.

One of the most famous uses of minimal criminals is in the proof of the four-colour theorem, where German geometer Heinrich Heesch realized in 1948 that if he could find a master set of patterns that can't appear in a minimal criminal, he would rule out all possible counter-examples and have a proof for it. These patterns were subsequently found using computer aid in 1976.