# Definition:Successor Set

Jump to navigation
Jump to search

## Contents

## Definition

Let $S$ be a set.

The **successor (set) of $S$** is defined and denoted:

- $S^+ := S \cup \set S$

## Also known as

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 $S'$ rather than $S^+$.

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

## Also see

## 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 $1.1$