Definition:Successor Set
(Redirected from Definition:Halmos Function)
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 $1.1$
