The Continuum Hypothesis

Tries to answer The Continuum Problem The size of is the smallest possible! In other words:

Theorem (Gödel)

We can’t refute the Continuum Hypothesis. More precisely, if Zermelo-Fraenkel Set Theory is consistent then

Proof

Follows from the Technique of Inner Models.

Theorem (Cohen)

We can’t prove the Continuum Hypothesis. More precisely, if Zermelo-Fraenkel Set Theory is consistent then