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# Def/Successor

Definition of Successor: Suppose that \$x\$ is a set. The successor of \$x\$ is the set \$x \cup \{ x \}\$ . We use the notation \$Succ(x)\$ for the successor of \$x\$.

Most frequently, \$Succ(x)\$ is used when \$x\$ is a natural number, or more generally, a ordinal number.

The axiom of regularity implies that: \$\$\forall x, Succ(x) \neq x.\$\$ Indeed, if \$x = x \cup \{ x \}\$, then \$x\$ would be an element of itself, contradicting the fact that no set is an element of itself.

## Logical Connections

This definition logically relies on the following definitions and statements: Def/Set, Def/Union, Def/Unordered pair, State/Axiom of regularity, State/No set is an element of itself

The following statements and definitions logically rely on the material of this page: Def/Ordinal number

To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic set theory, Clust/Cardinals and ordinals