Definition:Successor Mapping

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Let $V$ be a basic universe.

The successor mapping $s$ is the mapping on $V$ defined and denoted:

$\forall x \in V: \map s x := x \cup \set x$

where $x$ is a set in $V$.

Peano Structure

Let $\struct {P, s, 0}$ be a Peano structure.

Then the mapping $s: P \to P$ is called the successor mapping on $P$.

Successor Mapping on Natural Numbers

Let $\N$ be the set of natural numbers.

Let $s: \N \to \N$ be the mapping defined as:

$s = \set {\tuple {x, y}: x \in \N, y = x + 1}$

Considering $\N$ defined as a Peano structure, this is seen to be an instance of a successor mapping.

Successor Set

For $x \in V$, the result of applying the successor mapping on $x$ is denoted $x^+$:

$x^+ := \map s x = x \cup \set x$

$x^+$ is referred to as the successor (set) 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.

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.

Also see

  • Results about the successor mapping can be found here.