If X has $n$ elements then $\mathcal P$(X) has $2^n$ elements. Let X = {x,y,z}. By regularity, x≠{x}, and both x and {x} are subsets because everything is a set in axiomatic set theory. With this in mind, I'm struggling to understand why $\mathcal P$(X)={{},{x},{y},{z},{x,y},{x,z},{y,z},{x,y,z}} and not {{},x,y,z,{x},{y},{z},{x,y},{y,z},{x,z},{x,y,z}} unless we define x,y,z to be urelements. Is $\mathcal P$={{},{x},{y},{z},{x,y},{x,z},{y,z},{x,y,z}} because x, y, z are subsets that contain no elements and are therefore collectively represented by the empty set? Thanks for your time and help
[Math] The number of subsets in a Power Set
elementary-set-theory
Best Answer
It is true that in set theory we often take the case where there are no urelements, and everything is a set, but that doesn't mean that $x\in y$ implies $x\subseteq y$.
Consider the set $\{\{\{\varnothing\}\}\}$. It's a singleton, despite the fact that its element is a set. So it only has two subsets, $\varnothing$ and itself. So its power set would consist of these two elements. And despite the fact that $\varnothing$ is an element of an element of this set, it is not an element of the the set itself.