In
set theory
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, 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
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
ordinal other than
is additively indecomposable. Also,
is additively indecomposable, since the sum of two finite ordinals is still finite. More generally, every
infinite
Infinite may refer to:
Mathematics
* Infinite set, a set that is not a finite set
*Infinity, an abstract concept describing something without any limit
Music
*Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American m ...
initial ordinal
In a written or published work, an initial capital, also referred to as a drop capital or simply an initial cap, initial, initcapital, initcap or init or a drop cap or drop, is a letter at the beginning of a word, a chapter, or a paragraph that ...
(an ordinal corresponding to a
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
) 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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
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
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
), 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
In mathematics, Knuth's up-arrow notation is a method of notation for very large integers, introduced by Donald Knuth in 1976.
In his 1947 paper, R. L. Goodstein introduced the specific sequence of operations that are now called ''hyperoperati ...
.
See also
*
Ordinal arithmetic
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
References
*
{{PlanetMath attribution, urlname=additivelyindecomposable, title=Additively indecomposable
Ordinal numbers