Cardinality of the continuum

Lore & Background

The cardinality of the continuum was proven by Georg Cantor in his uncountability proof of 1874, part of his groundbreaking study of different infinities. Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if and only if there exists a bijective function between them. He showed that the real numbers are more numerous than the natural numbers, and that ℝ has the same number of elements as the power set of ℕ. The inequality was later stated more simply in his diagonal argument in 1891.

Between any two real numbers a < b, no matter how close, there are always infinitely many other real numbers, and Cantor showed that they are as many as those contained in the whole set of real numbers. The open interval (a,b) is equinumerous with ℝ, as well as with several other infinite sets, such as any n-dimensional Euclidean space ℝⁿ. The smallest infinite cardinal number is ℵ₀ (aleph-null), and the second smallest is ℵ₁ (aleph-one).

The continuum hypothesis asserts that there are no sets whose cardinality is strictly between ℵ₀ and 𝔠, meaning that 𝔠 = ℵ₁. This hypothesis is independent of the widely used Zermelo–Fraenkel set theory with axiom of choice (ZFC); that is, ZFC can neither prove that it is true nor that it is false.

Reader's Guide

The cardinality of the continuum is a foundational concept in set theory, introduced by Georg Cantor to compare the sizes of infinite sets. Cantor famously showed that the set of real numbers is uncountably infinite, meaning 𝔠 is strictly greater than ℵ₀. This result, proven in 1874 and later simplified via the diagonal argument in 1891, demonstrated that there are strictly more real numbers than integers. The cardinal equality 𝔠 = 2^ℵ₀ follows from the fact that the power set of ℕ has cardinality 2^ℵ₀, which is equal to 𝔠, as shown by one-to-one mappings in both directions between subsets of a countably infinite set and real numbers, applying the Cantor–Bernstein–Schroeder theorem. Cardinal arithmetic further shows that 𝔠² = 𝔠, and that 𝔠 = n·ℵ₀ for any finite cardinal n ≥ 2, as well as 𝔠 = ℵ₀·𝔠. The continuum hypothesis, which posits that 𝔠 = ℵ₁, remains independent of ZFC, meaning it can be neither proven nor disproven within that framework. This uncertainty underscores the depth and ongoing relevance of Cantor's work on the cardinality of the continuum.

Did You Know?

• The cardinality of the continuum is denoted by 𝔠 (lowercase Fraktur 'c') or |ℝ|.
• Cantor proved that the set of real numbers is uncountably infinite, meaning 𝔠 is strictly greater than ℵ₀.
• The open interval (a,b) is equinumerous with ℝ, as well as with any n-dimensional Euclidean space ℝⁿ.
• The continuum hypothesis, which asserts that 𝔠 = ℵ₁, is independent of ZFC set theory.