Continuum Hypothesis — Definition, Formula & Examples

The Continuum Hypothesis is the conjecture that there is no set whose cardinality is strictly between that of the natural numbers and that of the real numbers. In other words, it claims the real numbers form the "next size" of infinity after the countably infinite natural numbers.

The Continuum Hypothesis (CH) asserts that $2^{\aleph_0} = \aleph_1$ , meaning the cardinality of the power set of the natural numbers (equivalently, the cardinality of $\mathbb{R}$ ) equals the first uncountable cardinal. There is no cardinal $\kappa$ satisfying $\aleph_0 < \kappa < 2^{\aleph_0}$ .

Key Formula

2^{\aleph_0} = \aleph_1

Where:

$\aleph_0$ = The cardinality of the natural numbers (the smallest infinite cardinal)
$2^{\aleph_0}$ = The cardinality of the power set of the naturals, equal to the cardinality of the real numbers
$\aleph_1$ = The smallest cardinal strictly greater than \aleph_0

How It Works

Georg Cantor proved that

|\mathbb{R}| = 2^{\aleph_0} > \aleph_0

, so the reals are strictly "more infinite" than the naturals. The question then arises: is there some intermediate infinity between them? CH says no. Kurt Gödel (1940) showed CH is consistent with the standard axioms of set theory (ZFC), and Paul Cohen (1963) showed its negation is also consistent with ZFC. This means CH is independent of ZFC — it can be neither proved nor disproved from those axioms alone.

Example

Problem: Cantor showed |ℝ| > |ℕ|. Under the Continuum Hypothesis, what is the cardinality of ℝ expressed using aleph notation?

Step 1: Cantor's theorem gives us that the cardinality of the reals equals the cardinality of the power set of the naturals:

|\mathbb{R}| = 2^{\aleph_0}

Step 2: We know

2^{\aleph_0} > \aleph_0

. The next cardinal after

\aleph_0

in the aleph sequence is

\aleph_1

. The Continuum Hypothesis asserts there is nothing in between, so:

2^{\aleph_0} = \aleph_1

Answer: Under CH, the cardinality of ℝ is

\aleph_1

Why It Matters

The Continuum Hypothesis was the first problem on Hilbert's famous 1900 list of open problems. Its independence from ZFC, proved by Gödel and Cohen, revealed fundamental limits on what standard axioms can decide and launched the field of forcing in modern set theory. Students in logic, foundations of mathematics, or theoretical computer science encounter CH as a landmark example of an undecidable statement.

Common Mistakes

Mistake: Believing the Continuum Hypothesis has been proved or disproved.

Correction: CH is independent of ZFC: it is neither provable nor refutable from the standard axioms of set theory. Accepting or rejecting it requires adopting additional axioms.