Definition

Noun: - The smallest infinite cardinal number: In set theory, 'aleph-null' (symbolized as ℵ₀) is the cardinality (or size) of the set of all natural numbers (1, 2, 3, ...). It represents the smallest type of infinity, a countable infinity.

Usage

• As a mathematical term: Used exclusively in the context of set theory and transfinite mathematics to denote the cardinal number of countably infinite sets.
  • The set of all integers has the same cardinality, aleph-null, as the set of natural numbers.
  • Cantor proved that aleph-null is less than the cardinality of the real numbers.

Advanced Usage

• In comparisons of infinite sets: Used to discuss the hierarchy of infinities. A set is said to be "countably infinite" if its elements can be put into a one-to-one correspondence with the natural numbers, meaning its cardinality is aleph-null.
  • While both sets are infinite, the set of rational numbers is countably infinite (of size aleph-null), whereas the set of irrational numbers is not.

Variants and Related Words

• Aleph numbers (ℵ₀, ℵ₁, ℵ₂, ...): A sequence of numbers used to represent the cardinalities of infinite well-ordered sets. Aleph-null is the first in this sequence.
• Countable infinity: A descriptive term for an infinite set with cardinality ℵ₀.
• Cardinal number: A generalization of a natural number used to describe the size of a set, whether finite or infinite.

Synonyms

• ℵ₀ (the standard symbolic notation).
• Countable infinity (a conceptual synonym describing the type of infinity).

Related Concepts (Not Phrasal Verbs or Idioms)

• Continuum hypothesis: The proposition that there is no set whose cardinality is strictly between aleph-null and the cardinality of the real numbers (the continuum).
• One-to-one correspondence (bijection): The method used to establish that two sets have the same cardinality, including the cardinality aleph-null.

Noun

1. the smallest infinite integer