epsilon nought

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The correct title of this article is ε₀. It features superscript or subscript characters that are substituted or omitted because of technical limitations.

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

In mathematics, ε₀ is the smallest transfinite ordinal number which cannot be reached from 0 with a finite number of operations of ordinal addition and the operation α→ω^α, where ω is the smallest transfinite ordinal. As such it is a limit ordinal. It is given by

$\epsilon_0 = \omega^{\omega^{\omega^{\cdots}}},$

which stands for the limit of the sequence

$\omega, \omega^{\omega}, \omega^{\omega^{\omega}}, \omega^{\omega^{\omega^{\omega}}}, \cdots$

It has Cantor normal form

$\epsilon_0 = \omega^{\epsilon_0}.$

but this does not fully define it. The $γ$ -th ordinal (counting from zero) $α$ such that $α = ω α$ is written $\varepsilon_\gamma$ . These are called the epsilon numbers. The smallest of these numbers is ε₀.

The ordinal ε₀ 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 (as in Gentzen's consistency proof and the proof of Goodstein's theorem). Its use by Gentzen to prove the consistency of Peano arithmetic, along with Gödel's second incompleteness theorem, show that Peano arithmetic cannot prove the well-foundedness of this ordering (it is in fact the least ordinal with this property, and as such, in proof-theoretic ordinal analysis, is used as a measure of the strength of the theory of Peano arithmetic).

This was created by the German mathematician Georg Cantor. This ordinal is also called epsilon zero.

1 Rooted trees
2 Generalization
3 See also
4 References

[edit] Rooted trees

Finite rooted trees can be used to represent all ordinals less than ε₀ as follows. A finite rooted tree T represents the ordinal $\omega^{\alpha_1}+\cdots+\omega^{\alpha_n}$ where α₁≥....≥α_n are the ordinals represented by the n≥0 rooted trees obtained by deleting the root of T and the edges joined to it.

[edit] Generalization

If β is any ordinal such that $1 < β < ε α$ , then

$\beta^{\epsilon_\alpha} = \epsilon_\alpha$ .

As stated above,

$0, 1, \omega, \omega^{\omega}, \omega^{\omega^{\omega}}, \ldots \rightarrow \epsilon_0 \!$ .

Similarly, we get:

$\epsilon_0 + 1, \omega^{\epsilon_0 + 1}, \omega^{\omega^{\epsilon_0 + 1}}, \ldots \rightarrow \epsilon_1 \!$ and

$0, 1, \epsilon_0, \epsilon_0^{\epsilon_0}, \epsilon_0^{\epsilon_0^{\epsilon_0}}, \ldots \rightarrow \epsilon_1 \!$ .

More generally, for any ordinal $\alpha \!$ :

$\epsilon_{\alpha} + 1, \omega^{\epsilon_{\alpha} + 1}, \omega^{\omega^{\epsilon_{\alpha} + 1}}, \ldots \rightarrow \epsilon_{\alpha + 1} \!$ and

$0, 1, \epsilon_{\alpha}, \epsilon_{\alpha}^{\epsilon_{\alpha}}, \epsilon_{\alpha}^{\epsilon_{\alpha}^{\epsilon_{\alpha}}}, \ldots \rightarrow \epsilon_{\alpha + 1} \!$ .

Also when $\lambda \!$ is any limit ordinal:

$\epsilon_{\beta} \rightarrow \epsilon_{\lambda} \!$ as $\beta \rightarrow \lambda \!$ .

[edit] See also

Ordinal arithmetic

[edit] References

J.H. Conway, On Numbers and Games (1976) Academic Press ISBN 0-12-186350-6

v • d • e countable ordinals

ω · ε₀ · Feferman–Schütte ordinal Γ₀ · Ackermann ordinal · small Veblen ordinal · large Veblen ordinal · Bachmann–Howard ordinal · Church–Kleene ordinal ω₁^CK

Categories: Ordinal numbers

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epsilon nought

From Wikipedia, the free encyclopedia

Contents

[edit] Rooted trees

[edit] Generalization

[edit] See also

[edit] References

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