This article is about an ordinal in mathematics. For the physics constant ε₀, see permittivity.

In mathematics, ε₀ is the smallest transfinite ordinal number which cannot be reached from ω (the smallest transfinite ordinal) with a finite number of the ordinal operations of addition, multiplication and exponentiation. As such it is a limit ordinal. It is given by

$\backslash epsilon\_0\; =\; \backslash omega^\{\backslash omega^\{\backslash omega^\{\backslash cdots\}\}\},$

or in Cantor normal form by

$\backslash epsilon\_0\; =\; \backslash omega^\{\backslash epsilon\_0\}.$

The ordinal $\backslash epsilon\_0$ is still countable (there exist uncountable ordinals). This ordinal is very important in many induction proofs, because for many purposes, transfinite induction is only required up to ε₀.

This was created by the Russian-born mathematician Georg Cantor. It is also frequently cited by the Argentine-American mathematician and computer scientist Gregory Chaitin in his lectures and papers. This ordinal is also called "epsilon zero".

See also

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• ordinal arithmetic

External links

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• "A Century of Controversy over the Foundations of Mathematics" — A lecture given by Gregory Chaitin on April 30, 1999 at University of Massachusetts at Lowell.

Ordinal numbers