aleph-zero
Definition
Noun: - The smallest infinite cardinal number: In set theory, 'aleph-zero' (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
- 'Aleph-zero' is a technical term used primarily in mathematics, specifically in the field of set theory.
- It is used to describe and compare the sizes of infinite sets. A set whose elements can be put into a one-to-one correspondence with the natural numbers is said to have cardinality aleph-zero.
- Example:
- Example:
Advanced Usage
- "Countably infinite": This phrase is often used synonymously with having a cardinality of aleph-zero. A set is countably infinite if its elements can be listed in an infinite sequence.
- While the set of rational numbers (fractions) is dense, it is still countably infinite, or has size aleph-zero.
Variants and Related Words
- Aleph-null: This is another common name for aleph-zero.
- ℵ₀: The standard symbolic notation for aleph-zero.
- Countable infinity: A descriptive term for the concept represented by aleph-zero.
- Aleph numbers: A sequence of infinite cardinal numbers (ℵ₀, ℵ₁, ℵ₂, ...) of which aleph-zero is the first.
Synonyms
- Countable infinity (conceptual synonym)
- Aleph-null
Related Concepts (Not direct synonyms)
- Cardinality: The measure of the number of elements in a set.
- Continuum: Often refers to the cardinality of the real numbers, which is greater than aleph-zero.
- One-to-one correspondence (bijection): The method used to determine if two sets, including infinite ones, have the same cardinality.
Noun
-
the smallest infinite integer
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