The continuum hypothesis

We showed in Proposition 1 that $\aleph_0$ is the smallest infinite cardinal. It can be shown that there a next smallest cardinal called $\aleph_1$; i.e., the only cardinals strictly smaller than $\aleph_1$ are the finite ones and $\aleph_0$. Next come $\aleph_2$, $\aleph_3$, .... Where does ${\mathfrak{c}}=2^{\aleph_0} = \char93 {\mathbb{R}}$ fit into this list, if anywhere? (A priori, it could be bigger than $\aleph_n$ for every $n \in {\mathbb{N}}$.) We know that $2^{\aleph_0} \ge \aleph_1$, because we proved that $2^{\aleph_0}>\aleph_0$. Cantor conjectured

Continuum Hypothesis: $2^{\aleph_0} = \aleph_1$.

In other words, he believed that there is no set whose cardinality is strictly between that of ${\mathbb{N}}$ and that of ${\mathbb{R}}$.

In 1940 Gödel proved that the continuum hypothesis cannot be disproved from the other axioms of set theory. But in 1963 Cohen showed that it could not be proved from these axioms either!

The role of the continuum hypothesis in set theory is similar to the role of the parallel postulate in plane geometry. The parallel postulate (that given a line $L$ and a point $P$ not on $L$, there exists a unique line $L'$ through $P$ that does not intersect $L$) cannot be disproved from the other axioms of plane geometry, because it is actually true for the euclidean model of geometry. On the other hand, the parallel postulate cannot be proved either, since it is false in various noneuclidean models of geometry which do satisfy all the other axioms. Therefore the parallel postulate, or its negation, may be taken as a new axiom. Which one you choose will depend on your vision of what geometry is supposed to be.

Similarly, whether you choose to accept the continuum hypothesis will depend on your idea of what a set is supposed to be.