crabbo-rave / cantor

Proofs of Cantor's Theorems in Lean 3, which consequently prove that there are infinitely many infinities.

Proofs of Cantor's Theorems in Lean 3. They prove that $\mathcal{P}(A)$ is bigger than $A$ even if $A$ is infinite, and where $A$ is a set.

The Theorems

The two theorems proved were his proofs of Surjectivity and Injectivity. There are also some other small proofs scattered around.

Surjectivity

The theorem of Surjectivity states: Let $f$ be a map from set $A$ to its power set $\mathcal{P}(A)$. Then $f ~ : ~ A ~ \rightarrow ~ \mathcal{P}(A)$ is not surjective. As a consequence, $\text{card}(A) ~ < ~ \text{card}(\mathcal{P}(A))$ holds for any set $A$.

Here is a link to the handwritten proof.

What is extremely interesting about the theorem for Surjectivity is that it proves that there are infinitely many infinities, in an infinite heirarchy.

Injectivity

The thoerem of Injectivity states: Let $f$ be a map from power set $\mathcal{P}(A)$ to its set $A$. Then $f ~ : ~ \mathcal{P}(A) ~ \rightarrow ~ A$ is not injective.

Here is a link to the handwritten proof.

This proof is much less well known than the first, and it was hard to find a formlization online. I have to thank Neverbloom#6760 on discord for this one.

Glossary

powerset $\mathcal{P}$: Given a set $S$, $\mathcal{P}(S)$ is set of all subsets of $S$

surjective function: TODO