Conclude that there is no bijection between A and VA In part

Conclude that there is no bijection between A and V(A). In particular, conclude that V{N) is uncountable.

Solution

This is true by cantor\'s theorem.The theorem says that there is no surjection from a set A to power set P(A).The proof involves some language of axiomatic set theory so i am leaving that. Now as a bijection is also surjection hence proved. A set A is countable only if there is a bijection from set A to the set of natural numbers . but as we proved there is no such bijection possible as that means N has a bijection to P(N) which is not possible.