Another rendition of

Georg Cantor's proof with more modern notations.

**Theorem**
For any set A, the

power set P(A) is larger than A. This holds since there exists an obvious

injection from A to P(A), and not so obvious, there does not exist a

surjection from A to P(A).

**Proof by contradiction**
Suppose there exists

surjective f: A → P(A).

Let B = {a ∈ A : a !∈ f(a)}. (

"!∈" meaning "not in," for browsers that don't support ∉)

Since B ∈ P(A) and f

surjective, ∃ b ∈ A : f(b) = B.

But b exists

neither in B

nor not in B!

Contradiction. ∴ Surjective f cannot exist.

Russell was one of the people who asked about the case where A contained everything. This line of thought lead to the celebrated

Russell's Paradox. To resolve this

paradox, modern

axiomatic set theory was developed. (see:

ZF,

Axiom of Choice)