Aleph number

Lore & Background

The aleph numbers were introduced by Georg Cantor, who defined the notion of cardinality and realized that infinite sets can have different cardinalities. The smallest aleph number is ℵ₀ (aleph-nought, aleph-zero, or aleph-null), which is the cardinality of the set of natural numbers. A set has cardinality ℵ₀ if and only if it is countably infinite, meaning there is a bijection between it and the natural numbers. Examples include the set of all integers, rational numbers, algebraic numbers, and finite subsets of any countably infinite set.

The next aleph number, ℵ₁, is the cardinality of the set of all countable ordinal numbers, denoted ω₁. ℵ₁ is the smallest cardinality larger than ℵ₀. The definition of ℵ₁ implies (in ZF set theory without the axiom of choice) that no cardinal number lies between ℵ₀ and ℵ₁. If the axiom of choice is used, the class of cardinal numbers is totally ordered, making ℵ₁ the second-smallest infinite cardinal. Any countable subset of ω₁ has an upper bound in ω₁, analogous to finite sets of natural numbers having a maximum.

Continuing in this manner, it is possible to define an infinite cardinal number ℵ_α for every ordinal number α. The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus: alephs measure the sizes of sets, while infinity is often defined as an extreme limit of the real number line or an extreme point of the extended real number line.

Reader's Guide

The aleph numbers are significant because they provide a rigorous framework for understanding and comparing the sizes of infinite sets, a concept that revolutionized mathematics. By introducing ℵ₀, ℵ₁, and beyond, Georg Cantor showed that infinity is not a single concept but a hierarchy of distinct cardinalities. This work laid the foundation for modern set theory and influenced fields such as topology, analysis, and logic. The aleph numbers are central to the continuum hypothesis, which concerns the relationship between ℵ₁ and the cardinality of the real numbers. Their legacy endures in the study of infinite cardinalities, ordinal numbers, and the structure of the mathematical universe, challenging intuitive notions of size and infinity.

Did You Know?

• ℵ₀ is the cardinality of the set of natural numbers and is the smallest infinite cardinal.
• ℵ₁ is the cardinality of the set of all countable ordinal numbers, denoted ω₁.
• If the axiom of countable choice holds, ℵ₀ is smaller than any other infinite cardinal.
• The aleph numbers differ from the infinity (∞) used in algebra and calculus.