Cantor's diagonal argument
There's an enormous rabbit hole you can go down on various high level (at least for the layman) mathematical subjects, but this is one of my favourites. Most people get the concept of infinity and that it's really ****ing big, but kind of just assume it's some kind of theoretical constant or something like that. So to have the argument that's somewhat accessible to the standard bloke in terms of:
- we have this infinite set of stuff, it's called the numbers from 1, 2, 3 or however long you want to go
- we have this other infinite set of stuff, we'll call it every permutation of continual coin flips you can have - we'll map flip list 1 to the number 1, flip list 2 to the number 2, etc etc
- but wait, we can actually make a bigger infinite set by taking our list of flips and then swapping the first coin in list one from heads to tails or vice versa, the second coin in list two, the third coin in list three etc etc, and we now have an infinite set of stuff that is bigger than the other infinite set of stuff
It's like the concept of there being the same number of even numbers as there are natural numbers, except on steroids and in reverse.