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aleph

Represents the aleph number in set theory, denoting the cardinality of infinite sets.

Overview

Essential in mathematical set theory and foundational mathematics for discussing different sizes of infinity and transfinite numbers.

  • Most commonly used to represent ℵ₀ (aleph-null), the cardinality of natural numbers
  • Appears frequently in discussions of countable and uncountable sets
  • Important in advanced mathematics, particularly in topology and analysis
  • Used when discussing the continuum hypothesis and cardinal arithmetic

Examples

Expressing the cardinality of the set of natural numbers using aleph-null.

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

Comparing the cardinality of real numbers to aleph-null.

∣R∣=2ℵ0>ℵ0\left|\mathbb{R}\right| = 2^{\aleph_0} > \aleph_0 
\left|\mathbb{R}\right| = 2^{\aleph_0} > \aleph_0

Stating the continuum hypothesis using aleph notation.

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