# Definition:Successor Mapping/Also known as

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## 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*.

Some sources use $x'$ rather than $x^+$.

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 - 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