Additively indecomposable ordinal explained

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any

\beta,\gamma<\alpha , we have \beta + \gamma<\alpha. Additively indecomposable ordinals were named the gamma numbers by Cantor,^{[1] p.20} and are also called additive principal numbers. The class of additively indecomposable ordinals may be denoted H , from the German "Hauptzahl".^[2] The additively indecomposable ordinals are precisely those ordinals of the form \omega ^{\beta} for some ordinal \beta .

From the continuity of addition in its right argument, we get that if

\beta < \alpha and α is additively indecomposable, then \beta + \alpha = \alpha.

Obviously 1 is additively indecomposable, since

0+0 < 1 . No finite ordinal other than 1 is additively indecomposable. Also, \omega is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every infinite initial ordinal (an ordinal corresponding to a cardinal number) is additively indecomposable.

The class of additively indecomposable numbers is closed and unbounded. Its enumerating function is normal, given by

\omega ^{\alpha} .

The derivative of

\omega ^{\alpha} (which enumerates its fixed points) is written \varepsilon _{\alpha} Ordinals of this form (that is, fixed points of \omega ^{\alpha} ) are called epsilon numbers. The number \varepsilon _{0} = \omega ^{\omega ^{\omega ^{…}}} is therefore the first fixed point of the sequence \omega,\omega ^{\omega,\omega} ^{\omega ^{\omega}} ,\ldots

Multiplicatively indecomposable

A similar notion can be defined for multiplication. If α is greater than the multiplicative identity, 1, and β < α and γ < α imply β·γ < α, then α is multiplicatively indecomposable. The finite ordinal 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (named the delta numbers by Cantor^p.20) are those of the form

\omega ^{\omega ^{\alpha}} for any ordinal α. Every epsilon number is multiplicatively indecomposable; and every multiplicatively indecomposable ordinal (other than 2) is additively indecomposable. The delta numbers (other than 2) are the same as the prime ordinals that are limits.

Higher indecomposables

Exponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of

\varepsilon _{\alpha} ), and so on. Therefore, \varphi _{\omega(0)} is the first ordinal which is \uparrow ^{n} -indecomposable for all n , where \uparrow denotes Knuth's up-arrow notation.

See also

• Ordinal arithmetic

Notes and References

1. A. Rhea, "The Ordinals as a Consummate Abstraction of Number Systems" (2017), preprint.
2. W. Pohlers, "A short course in ordinal analysis", pp. 27–78. Appearing in Aczel, Simmons, Proof Theory: A selection of papers from the Leeds Proof Theory Programme 1990 (1992). Cambridge University Press,