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beth

Represents the beth numbers in set theory and mathematical logic, used to denote specific infinite cardinal numbers.

Overview

Essential in advanced set theory for representing beth numbers, which form an important sequence of cardinal numbers defined using the power set operation.

  • Commonly appears in discussions of the continuum hypothesis and cardinal arithmetic
  • Used in mathematical logic and foundations of mathematics
  • Often encountered alongside other Hebrew letter symbols in set-theoretic contexts
  • Particularly relevant when discussing infinite sets and their relative sizes

Examples

Defining a cardinal number using beth notation in set theory.

ℵ0<ℶ1=2ℵ0\aleph_0 < \beth_1 = 2^{\aleph_0} 
\aleph_0 < \beth_1 = 2^{\aleph_0}

Comparing beth numbers in cardinal arithmetic.

ℶ0<ℶ1<ℶ2<ℶ3\beth_0 < \beth_1 < \beth_2 < \beth_3 
\beth_0 < \beth_1 < \beth_2 < \beth_3

Expressing the continuum hypothesis using beth notation.

2ℵ0=ℶ1=ℵ12^{\aleph_0} = \beth_1 = \aleph_1 
2^{\aleph_0} = \beth_1 = \aleph_1