In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
, the successor of an
ordinal number
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets.
A finite set can be enumerated by successively labeling each element with the leas ...
''α'' is the smallest ordinal number greater than ''α''. An ordinal number that is a successor is called a successor ordinal. The ordinals 1, 2, and 3 are the first three successor ordinals and the ordinals ω+1, ω+2 and ω+3 are the first three infinite successor ordinals.
Properties
Every ordinal other than 0 is either a successor ordinal or a
limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
.
[.]
In Von Neumann's model
Using
von Neumann's ordinal numbers (the standard model of the ordinals used in set theory), the successor ''S''(''α'') of an ordinal number ''α'' is given by the formula
:
Since the ordering on the ordinal numbers is given by ''α'' < ''β'' if and only if ''α'' ∈ ''β'', it is immediate that there is no ordinal number between α and ''S''(''α''), and it is also clear that ''α'' < ''S''(''α'').
Ordinal addition
The successor operation can be used to define
ordinal addition rigorously via
transfinite recursion
Transfinite induction is an extension of mathematical induction to well-ordered sets, for example to sets of ordinal numbers or cardinal numbers. Its correctness is a theorem of ZFC.
Induction by cases
Let P(\alpha) be a property defined for a ...
as follows:
:
:
and for a limit ordinal ''λ''
:
In particular, . Multiplication and exponentiation are defined similarly.
Topology
The successor points and zero are the
isolated points of the class of ordinal numbers, with respect to the
order topology.
[.]
See also
*
Ordinal arithmetic
*
Limit ordinal
In set theory, a limit ordinal is an ordinal number that is neither zero nor a successor ordinal. Alternatively, an ordinal λ is a limit ordinal if there is an ordinal less than λ, and whenever β is an ordinal less than λ, then there exists a ...
*
Successor cardinal In set theory, one can define a successor operation on cardinal numbers in a similar way to the successor operation on the ordinal numbers. The cardinal successor coincides with the ordinal successor for finite cardinals, but in the infinite case th ...
References
{{DEFAULTSORT:Successor Ordinal
Ordinal numbers