202405240908
Status: #idea
Tags: Set Theory
State: #awakened
Axiom Of Choice
It states that:
A Cartesian product of nonempty sets is nonempty.
In other words, given a (possibly infinite) collection of nonempty sets, we can make a selection from each of the sets all at once. It does not specify how to do it, but it stipulates that we always can. It is an extremely natural axiom that arises from our intuition in the real world, but surprisingly it is not necessary. It is necessary in the sense that almost all of modern math is built on it, and almost every mathematician uses it even if they are not aware of it.
But akin to non-euclidean geometry and the idea that there are configurations under which parallel lines may cross, similarly their are systems of math which refuse the axiom of choice and end up with entirely new systems of mathematic that are orthogonal to the one we are used to. In practice, I don't know how useful they are, but they exist.
It's a technique that is used Set Theory to in a sense put the onus on the reader to prove something. It allows me to state that things with some properties exist without me having to prove that they do exist since as long as I can say that there is a way to sample something, then the property I want follows.