When defining the ordinal numbers, an absolutely fundamental operation that we can perform on them is a successor operation S to get the next higher one. Using von Neumann's ordinal numbers (the standard ordinals used in set theory), we have, for any ordinal number,

$S(\backslash alpha)\; =\; \backslash alpha\; \backslash cup\; \backslash \{\backslash alpha\backslash \}.$

Since the ordering on the ordinal numbers α < β if and only if $\backslash alpha\; \backslash in\; \backslash beta$, it is immediate that there is no ordinal number between α and S(α) and it is also clear that α < S(α). An ordinal number which is S(β) for some ordinal β is called a successor ordinal. Ordinals which are neither zero nor successors are called limit ordinals. We can use this operation to define ordinal addition rigorously via transfinite recursion as follows:

$\backslash alpha\; +\; 0\; =\; \backslash alpha$

$\backslash alpha\; +\; S(\backslash beta)\; =\; S(\backslash alpha\; +\; \backslash beta)$

and for a limit ordinal λ

In particular, S(α) = α + 1. Multiplication and exponentiation are defined similarly.

See also

• ordinal arithmetic
• limit ordinal
• Successor cardinal

Ordinal numbers

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