Aleph one, written ℵ1, names the smallest infinite cardinal that is uncountable. Cardinal numbers measure sizes of sets beyond finite size; ℵ1 is the first such size strictly larger than aleph null (ℵ0).
More precisely, ℵ1 is the cardinality of the set of all countable ordinals, commonly denoted ω1. That set itself is uncountable, and its cardinality is by definition the least uncountable cardinal. The term "cardinality" is used in a technical sense; for an introduction see cardinality.
Key characteristics
- Successor: ℵ1 is the next aleph after ℵ0; it is the first uncountable aleph.
- Initial ordinal: its corresponding initial ordinal is ω1, the first uncountable ordinal.
- Regularity: ℵ1 has uncountable cofinality and is a regular cardinal in standard set theory.
Historically the aleph notation comes from Georg Cantor's work on transfinite numbers. Later developments in 20th-century set theory clarified relationships between alephs and the continuum (the size of the real line). Two landmark results showed that whether the cardinality of the continuum equals ℵ1—the continuum hypothesis—cannot be decided from the usual axioms of Zermelo–Fraenkel set theory with Choice: both its consistency and its independence were established by Gödel and Cohen in different senses.
In practice ℵ1 appears across mathematical topics as a threshold between "countable" and "uncountable". In topology and analysis it often signals new phenomena: certain constructions behave differently once uncountable sets are involved. In pure set theory ℵ1 is central to questions about cardinal arithmetic, forcing, and models of set theory.
A key distinction to remember is that ℵ1 need not equal the cardinality of the real numbers (the continuum). The continuum is denoted by c and equals 2^{ℵ0}; the continuum hypothesis asserts c = ℵ1, i.e. that the reals have cardinality ℵ1, but this claim is independent of the usual axioms of set theory. For discussion of the reals see real numbers. Finally, ℵ1 is followed by ℵ2, the next larger aleph.