# Definition:Successor

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## Disambiguation

This page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article.

**Successor** may refer to:

- Successor element: In an ordered set $\left({S, \preceq}\right)$, $a$ is a successor element to $b$ if and only if $b \prec a$.

- Immediate successor element: In an ordered set $\left({S, \preceq}\right)$, $a$ is the immediate successor element to $b$ if and only if $b \prec a$ and $\nexists c \in S: b \prec c \prec a$.

- Successor set: If $S$ is a set, then its successor set $S^+$ is defined as $S^+ := S \cup \left\{{S}\right\}$.

- Successor mapping: The mapping at the heart of a Peano structure which encapsulates its ability to sustain the Principle of Mathematical Induction.

- Successor ordinal: An ordinal which is the successor set of some other ordinal.