In this thread we collect and talk about random bits of science that make us horny.
Make sure it's bizarre but also try to make it accessible!
I'll go first with some interrelated bits of set theory:
- It's possible to make a one-to-one mapping between the points of n dimensional space and m dimensional space. For example, using this idea, you can fill a 3D room with an infinitely thin 1D string, with no gaps.
- The set of everything, or the set of all possible sets, cannot exist, as it creates a paradox. This is because, in general, a set is smaller than the set of all of its subsets. However, the set of everything must contain, among other things, the set of all its subsets (if it didn't it wouldn't be the set of everything!), hence it is LARGER than the set of its subsets at the same time as being SMALLER.
- There are different sizes of infinity. The smallest type is the infinity that represents all ordinary counting numbers; 1, 2, 3, ... . This is the same size as the infinity of positive and negative numbers together, and also fractions.
The awesome fact is that the infinity that counts the continuous numbers (all of the decimals) is BIGGER than the previous infinity.
However, it is impossible to determine whether there is an infinity between the two. This means that maths has split into two different sets of truths: one where you can assume that there is a middling infinity, and one where you assume there isn't.
Above the continuous infinity, there are even bigger infinities. In fact, they go on infinitely.
If you try to count how big the infinity of infinities is, though, you get another paradox. :D