hey @socks, thought of a math thing that seems like the kind of thing you would know
given an infinite space (let's say the 2D plane, or the field of complex numbers if that makes it easier), is it in general possible to define a one-to-one relationship between points in that space and sets of points in the same space? Such that every possible point corresponds to exactly one set, and every possible set corresponds to exactly one point
-F
@socks this thought brought to you by thinking about defining gender as a space, and whether one's gender would be a single point, or a region, or even multiple disjoint regions
-F
@Felthry I see you already got some responses, but yeah, the answer is no by Cantor's theorem!
Having a bijection (a one-to-one relation) between sets is the same as those sets having the same cardinality. That's kinda what cardinality IS in a sense. And Cantor's theorem states that the power set (the set of all subsets) always has a greater cardinality than the set.
So not only is it impossible in general, it's ALWAYS impossible! Even for the empty set, which has zero elements, and its power set has one.
@Felthry That would fix the specific case where the set's power set happens to have the same cardinality as the real line, which is only true for infinite countable sets.
An infinite countable set has the same cardinality as the integers, and its power set has the same cardinality as the real line. If you want such a bijection, intuitively, you could chop up the real line into segments of length 1, each of one also has the same cardinality as the real line. So, you can have a bijection between the real line and one of those segments, and assign each one to an element of the set.