Definition:Successor Mapping/Peano Structure
Definition
Let $\struct {P, s, 0}$ be a Peano structure.
Then the mapping $s: P \to P$ is called the successor mapping on $P$.
Successor Element
The image element $\map s x$ of an element $x$ is called the successor element or just successor 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
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 4$: The natural numbers
- 1964: J. Hunter: Number Theory ... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $2$. The positive integers
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 1$: Introduction