Definition:Successor Mapping/Successor Set

Definition

Let $V$ be a basic universe.

Let $s: V \to V$ denote the successor mapping on $V$.

For $x \in V$, the result of applying the successor mapping on $x$ is denoted $x^+$:

$x^+ := \map s x = x \cup \set x$

$x^+$ is referred to as the successor (set) of $x$.

Also known as

The successor mapping can also be seen referred to as the successor function.

Some sources call this the Halmos function, for Paul R. Halmos who made extensive use of it in his $1960$ work Naive Set Theory.

Various different notations can be found:

$\map s x$

$\map S x$

$x^+$

$x'$

Some sources use $x + 1$ rather than $x^+$, on the grounds that these coincide for the natural numbers (when they are seen as elements of the von Neumann construction of natural numbers).

Smullyan and Fitting, in their Set Theory and the Continuum Problem, revised ed. of $2010$, use a variant of $\sigma$ which looks like $o$ with $^\text {-}$ as a close superscript.

Also see

• Results about the successor mapping can be found here.

Sources

• 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 11$: Numbers
• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.22$
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $3$: The Natural Numbers: $\S 1$ Preliminaries: Definition $1.1$