Definition:Successor Mapping/Also known as

Successor Mapping: 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.

Sources

• 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Peano's postulates
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Peano's postulates
• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $5$: Ordinal Numbers: $\S 1$ Ordinal numbers