Definition

1. Noun:
   • The smallest infinite cardinal number: In set theory, 'aleph-nought' (also written as ℵ₀) is the cardinality of the set of all natural numbers. It represents the smallest size of infinity, specifically the infinity of countable sets.

Usage

• 'Aleph-nought' is a technical term used almost exclusively in the field of mathematics, particularly in set theory and discussions of infinity.
• It is used to denote the cardinal number associated with countably infinite sets, such as the set of integers or rational numbers.
• Example:
• Example:

Advanced Usage

• "Aleph-null": An alternative and equally common name for 'aleph-nought'.
  • The cardinality of the natural numbers is aleph-null.
• In comparisons of infinities: Used to establish that other infinities (like the continuum, the cardinality of the real numbers) are larger than ℵ₀.
  • The continuum hypothesis concerns the relationship between aleph-nought and the cardinality of the real numbers.

Variants and Related Words

• Aleph-null (n): A direct synonym for aleph-nought.
• Aleph-zero (n): Another less common variant.
• Countable infinity (n): A descriptive phrase for the concept represented by aleph-nought.
• ℵ₀ (symbol): The standard symbolic notation for aleph-nought.

Synonyms

• Aleph-null
• Aleph-zero
• Countable infinity (conceptual synonym)

Related Concepts (Not Phrasal Verbs or Idioms)

• Cardinal number: A generalization of a natural number used to describe the size of a set.
• Countable set: A set with the same cardinality as some subset of the natural numbers; its size is at most aleph-nought.
• Uncountable set: A set, like the real numbers, whose cardinality is greater than aleph-nought.

Noun

1. the smallest infinite integer