The Continuum

The size of the real line has a name: the continuum, written \mathfrak{c}. We know it is bigger than \aleph_0 because the reals are uncountable. But Cantor's theorem pins down exactly how big: the continuum is the size of the power set of the naturals,

\mathfrak{c} = |\mathbb{R}| = |\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}.

The reason is a clean coding: every real in [0,1] is an infinite binary expansion 0.b_1 b_2 b_3 \dots, and each such sequence of 0s and 1s picks out a subset of \mathbb{N} (put n in the set exactly when b_n = 1). Reals and subsets of \mathbb{N} are the same in number: 2^{\aleph_0}.

Is there anything in between?

We now have two infinities in hand, \aleph_0 < \mathfrak{c}, and a natural question: is there a set whose size is strictly between them — bigger than the naturals, but smaller than the reals? Cantor believed the answer was no, that \mathfrak{c} is the very next infinity after \aleph_0. That conjecture is the Continuum Hypothesis (CH):

Statement: there is no set with cardinality strictly between \aleph_0 and \mathfrak{c} = 2^{\aleph_0} — equivalently \mathfrak{c} = \aleph_1, the next cardinal up.
Verdict: CH is independent of the standard axioms (ZFC) — neither provable nor disprovable from them.
Gödel (1940) showed CH cannot be disproved; Cohen (1963) showed it cannot be proved, inventing the method of forcing to do so.

This is one of the most astonishing results in mathematics: a perfectly precise question about the sizes of number sets has no answer within our usual foundations. You can add CH as an axiom, or add its negation, and either way you get a consistent mathematics. It was the very first problem on Hilbert's famous 1900 list.

Here is a fact that unsettled even Cantor. The plane \mathbb{R}^2 — all pairs of reals — has the same cardinality as the line \mathbb{R}: |\mathbb{R}^2| = \mathfrak{c}. You can interleave the digits of two coordinates into a single real and back again, pairing the plane with the line. Cantor wrote to a colleague, "I see it, but I don't believe it." Dimension, it turns out, doesn't change the size of the continuum: |\mathbb{R}| = |\mathbb{R}^2| = |\mathbb{R}^n| = \mathfrak{c} for every finite n.

\mathfrak{c} = 2^{\aleph_0} is a theorem. But "\mathfrak{c} = \aleph_1" (the Continuum Hypothesis) is unprovable — don't assume the continuum is the next cardinal after \aleph_0; that's exactly the undecidable part.
Cantor's theorem still guarantees the ladder keeps climbing: \mathfrak{c} < 2^{\mathfrak{c}} < \cdots. Independence of CH is about the gap below \mathfrak{c}, not about running out of bigger infinities.