Axiom of Fusion
Imagine an infinitely sharp knife capable of cutting an interval exactly in the middle. We will call the knife, deknife in honor of Dedekind. Imagine deknife starting with a unit interval and indefinitely continuing the cutting with the resulting pieces. It is easy to see that every infinite sequence of deknife cutting can be represented by a binary number in the unit interval. The interesting question is, if all infinite sequences of deknife cutting is allowed, what is the length of the resulting pieces. Surely, it cannot be zero. Axiom of Fusion tells you what it is. But again we need sophisticated notations.