• Source: Successor ordinal

In set theory, the successor of an ordinal number 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 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




S
(
α
)
=
α
∪
{
α
}
.


{\displaystyle S(\alpha )=\alpha \cup \{\alpha \}.}


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 as follows:




α
+
0
=
α



{\displaystyle \alpha +0=\alpha \!}





α
+
S
(
β
)
=
S
(
α
+
β
)


{\displaystyle \alpha +S(\beta )=S(\alpha +\beta )}


and for a limit ordinal λ




α
+
λ
=

⋃

β
<
λ


(
α
+
β
)


{\displaystyle \alpha +\lambda =\bigcup _{\beta <\lambda }(\alpha +\beta )}


In particular, S(α) = α + 1. 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
Successor cardinal




References

Kata Kunci Pencarian:

• Yūichirō Umehara
• Successor ordinal
• Ordinal number
• Successor cardinal
• Limit ordinal
• Ordinal arithmetic
• Successor
• Aleph number
• Order topology
• Limit cardinal
• Veblen function