Additively indecomposable ordinal

In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any ${\displaystyle \beta ,\gamma <\alpha }$, we have ${\displaystyle \beta +\gamma <\alpha .}$ Additively indecomposable ordinals are also called gamma numbers. The additively indecomposable ordinals are precisely those ordinals of the form ${\displaystyle \omega ^{\beta }}$ for some ordinal ${\displaystyle \beta }$.

From the continuity of addition in its right argument, we get that if ${\displaystyle \beta <\alpha }$ and α is additively indecomposable, then ${\displaystyle \beta +\alpha =\alpha .}$

Obviously 1 is additively indecomposable, since ${\displaystyle 0+0<1.}$ No finite ordinal other than ${\displaystyle 1}$ is additively indecomposable. Also, ${\displaystyle \omega }$ 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 ${\displaystyle \omega ^{\alpha }}$.

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

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. 2 is multiplicatively indecomposable since 1·1 = 1 < 2. Besides 2, the multiplicatively indecomposable ordinals (also called delta numbers) are those of the form ${\displaystyle \omega ^{\omega ^{\alpha }}\,}$ 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.

See also

• Ordinal arithmetic

References

• Sierpiński, Wacław (1958), Cardinal and ordinal numbers, Polska Akademia Nauk Monografie Matematyczne, 34, Warsaw: Państwowe Wydawnictwo Naukowe, MR 0095787

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