Continuum Hypothesis¶

Definition¶

The continuum hypothesis is the hypothesis that there is no set whose cardinality is strictly between $\aleph_0$ and $\mathfrak{c}$.

Interpretation¶

Aleph null, $\aleph_0$, is the size of the natural numbers; the cardinality of the continuum, $\mathfrak{c}$, is the size of the reals. These are the two endpoints between which the hypothesis rules out an intermediate cardinality.

It asks whether an intermediate cardinality lies between those of the natural numbers and the real numbers.

Assuming the axiom of choice, the continuum hypothesis is equivalent to $\mathfrak{c}=\aleph_1$${\aleph }_1$, where $\aleph_1$ is the smallest cardinality larger than $\aleph_0$. Assuming the standard axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC) are consistent, they can neither prove nor refute this hypothesis (Gödel, 1940; Cohen, 1963).

Key Results¶

• In symbols, it denies the existence of a set $A$ such that $\aleph_0<\operatorname{card}(A)<\mathfrak{c}$$\mathfrak{c}$.
• A bijection is the main tool for proving that two sets have the same cardinality.

Examples¶

• The question compares the size of $\mathbb{N}$ with the size of $\mathbb{R}$.

Prerequisites¶

• Aleph Null
• Cardinality of the Continuum

Related Concepts¶

• Power Set
• Countable and Uncountable Sets
• Cantor's Theorem
• Cantor's Diagonal Argument

Sources¶

• Math II: Linear Algebra · Week 1

Aliases: CH, Cantor's continuum hypothesis Symbols: natural numbers, real numbers

Last updated: 27 October 2026 11:52