Successor of a set

Note: successor function redirects here, it is certainly a synonym but certainly not the best name

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Definition

Let $X$ be a set. The successor of the set $X$, written $S(x)$, is defined as follows:

• $S(x):\eq x\cup \{x\}$

Claim 1: such a set exists

Terminology

I prefer and use:

• "successor of $x$"

But not as one might read "$f(x)$" as "$f$ of $x$". At the point which this is usually defined (before the Axiom of infinity) - even if relations are covered, and thus functions are defined, we cannot phrase this as a function.

Proof of claims

• Let $x$ be a given set
  • By the Axiom of paring $\exists A\forall a[a\in A\iff(a\eq x\vee a\eq x)]$ - where equality is understood as per the Axiom of extensionality
    • The paring is unique by extensionality. The $A$ posited to exist is written as $\{x\}$ (we have no concept of a singleton yet, this is notation for {{M|\{x,x\} ]} - a pair of $x$s)
  • By the axiom of paring again: $\exists B\forall b[b\in B\iff(b\in x\vee b\in \{x\})]$
    • the $B$ posited to exist is written $\{x,\{x\}\}$
  • By the Axiom of union: $\exists C\forall c[c\in C\iff\exists D\in\{x,\{x\}\}(c\in D)]$
    • We denote the $C$ posited to exist by $\bigcup\{x,\{x\}\}$ (or as a slight abuse of notation at this point: $x\cup\{x\}$ - as required)

See also

• Inductive set
• The natural numbers - denoted $\mathbb{N}$