The Continuum Hypothesis, often abbreviated $\mathrm{CH}$, is a statement about the size of the cardinality of the continuum $\mathfrak{c}=2^{\aleph_0}$ relative to other cardinals in ZF set theory . In particular, it states that $\aleph_1=2^{\aleph_0}$, or that the cardinality of the real numbers is the smallest cardinality strictly greater than the cardinality of countable sets.
It happens that $\mathrm{CH}$ is independent from the axioms of ZFC . Proving this is very difficult and requires a lot of model-theoretic machinery, such as a technique called forcing, but the following is a rough sketch of the "game plan" for proving this:
The LĂ¶wenheim-Skolem-Tarski Theorems can be used to establish the existence of countable transitive models $\mathrm{M}$ of ZFC. These are used as generic "base models" from which more exotic models are constructed.
By virtue of the countability of $\mathrm{M}$, it is necessarily lacking some of the sets of the "fuller" model in which it is contained. Using the constructions of forcing notions and the P-name hierarchy , we are able to define generic extension models $\mathrm{M}[G]$ consisting of $\mathrm{M}$ "augmented by" $G$ and some additional sets necessary to ensure that $\mathrm{M}[G]$ still satisfies the axioms of ZFC.
If $\mathrm{CH}$ is not satisfied in some model $\mathrm{M}$, then there exists no bijection between $2^\mathbb{N}$ and $\aleph_1$ in $M$. However, since $\mathrm{M}$ is countable there does exist a bijection $f:2^\mathbb{N}\to\aleph_1^M$ - it just isn't included in $\mathrm{M}$. By choosing an appropriate forcing notion $P$ and generic ideal $G$, we can augment $\mathrm{M}$ to form a model $\mathrm{M}[G]$ which does include $f$. Hence, we have a model in which $\mathrm{CH}$ holds.
On the other hand, if $\mathrm{CH}$ holds in $\mathrm{M}$, we can add a different kind of set to $\mathrm{M}$ to ensure that $2^\mathbb{N}\geq\aleph_2$ and $\mathrm{CH}$ fails in $\mathrm{M}[G]$. In particular, we can do this by adding a set of new distinct subsets of $2^\mathbb{N}$ to $\mathrm{M}$ - one for each element of $\aleph_2^M$. (Some additional work must be done to make sure that we don't "collapse any cardinals", or inadvertently destroy the strict inequalities $2^\mathbb{N}<\aleph_1^M$ and $\aleph_1 < \aleph_2$ in the process of adjoining $G$.)