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alef

Represents the Hebrew letter Alef, commonly used in set theory to denote infinite cardinal numbers.

Overview

A fundamental symbol in mathematical set theory and logic, particularly important for discussing different sizes of infinity and transfinite numbers.

  • Central to Cantor's theory of infinite sets
  • Used to denote the cardinality of countably infinite sets
  • Appears frequently in advanced mathematics, especially when discussing the continuum hypothesis
  • Often encountered in abstract algebra and topology when dealing with infinite structures

Examples

Defining the cardinality of the set of natural numbers.

ℵ0=∣N∣\aleph_0 = |\mathbb{N}| 
\aleph_0 = |\mathbb{N}|

Comparing cardinalities of infinite sets.

∣R∣=ℵ1>ℵ0|\mathbb{R}| = \aleph_1 > \aleph_0 
|\mathbb{R}| = \aleph_1 > \aleph_0

Expressing the continuum hypothesis.

2ℵ0=ℵ12^{\aleph_0} = \aleph_1 
2^{\aleph_0} = \aleph_1