There are lots of things we don’t know; personally I’m a veritable cornucopia of ignorance. But there is a difference between things we don’t know and things that can’t be known. For example, no-one knows when Shakespeare was born (although we do know when he was baptized). However, it’s not impossible that in the future we could find out – a long lost document might be found that mentions his birth, so Shakespeare’s true date of birth is not unknowable, just unknown. This list contains 10 things that are unknowable in principle. Not only are they unknown now, they can never be known.
Most of these are mathematical; I’ve tried to make it as nontechnical as possible – apart from anything else, I’m no mathematician so I’ve tried to dumb it down enough so that I can understand it.
Unknowable Thing: What’s in a set of sets that don’t contain themselves?
We have to do a little mathematics for several of these items! This is the first on the list because, in a sense, the concept of the “unknowable” starts with this paradox discovered by Bertrand Russell in 1901.
Let’s start with the idea of a set. A set is a collection of objects – for example, you could have the set of positive even numbers that contains 2, 4, 6, 8… or the set of prime numbers containing 2, 3, 5, 7, 11… so far so good.
Can sets contain other sets? Yes, no problem – you could have a set of sets that contain other sets – and that set would, obviously, contain itself. In fact, you can split sets into two types – those that contain themselves and those that don’t.
So, consider a set (S, say) of sets that don’t contain themselves. Does S contain itself? If it does, then it shouldn’t be in the set, but if it doesn’t, then it should. So S is continually hopping in and out of itself.
This paradox caused quite a lot of consternation amongst mathematicians. Imagine someone proving that a number could be simultaneously even and odd, it’s similarly worrisome to that. Ways have been gotten around the paradox – essentially by redefining set theory.
