THE DIVINE RANDOM
Omega Numbers
The harder we stare, the less we see
A simple working definition of randomness, that it be universally unpredictable, can be made more precise in the realm of mathematics. Namely, something is mathematically random if there exists no axiomatic system (with a finite set of axioms) from which it can be derived. A random number, in particular, is a number whose digits cannot be predicted within the framework of mathematics. Such a number is irreducible — the shortest possible way to express one is to write down all its (infinitely many) digits. If we can agree that a name for something is just a finite sequence of symbols that represent it, these numbers are necessarily nameless. Even numbers like π and e, which are transcendental, their digits never repeat, have names — π and e are the names we have given to two specific infinite real numbers that can be computed in two very specific, and finite, ways. Mathematically random (aka irreducible) numbers are even weirder than transcendentals. So weird that it becomes reasonable to ask, do such numbers actually exist, and if so, can you show me one?
Do random numbers actually exist?
It’s actually pretty easy to see that irreducible numbers, numbers that cannot be derived within any Formal Axiomatic System (FAS), do indeed…
