Additively indecomposable ordinal

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In set theory, a branch of mathematics, an additively indecomposable ordinal α is any ordinal number that is not 0 such that for any , we have Additively indecomposable ordinals are also called gamma numbers or additive principal numbers. The additively indecomposable ordinals are precisely those ordinals of the form for some ordinal .

From the continuity of addition in its right argument, we get that if and α is additively indecomposable, then

Obviously 1 is additively indecomposable, since No finite ordinal other than is additively indecomposable. Also, 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 .

The derivative of (which enumerates its fixed points) is written Ordinals of this form (that is, fixed points of ) are called epsilon numbers. The number is therefore the first fixed point of the sequence

Multiplicatively indecomposable[edit]

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 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[edit]

Exponentially indecomposable ordinals are equal to the epsilon numbers, tetrationally indecomposable ordinals are equal to the zeta numbers (fixed points of ), and so on. Therefore, the Feferman-Schutte ordinal (fixed point of ) is the first ordinal which is -indecomposable for all , where denotes Knuth's up-arrow notation.

See also[edit]


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

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