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 [ilmath]X[/ilmath] be a set. The successor of the set [ilmath]X[/ilmath], written [ilmath]S(x)[/ilmath], is defined as follows:

• [ilmath]S(x):\eq x\cup \{x\} [/ilmath]

Claim 1: such a set exists

Terminology

I prefer and use:

• "successor of [ilmath]x[/ilmath]"

But not as one might read "[ilmath]f(x)[/ilmath]" as "[ilmath]f[/ilmath] of [ilmath]x[/ilmath]". 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 [ilmath]x[/ilmath] be a given set
  • By the Axiom of paring [ilmath]\exists A\forall a[a\in A\iff(a\eq x\vee a\eq x)][/ilmath] - where equality is understood as per the Axiom of extensionality
    • The paring is unique by extensionality. The [ilmath]A[/ilmath] posited to exist is written as [ilmath]\{x\} [/ilmath] (we have no concept of a singleton yet, this is notation for {{M|\{x,x\} ]} - a pair of [ilmath]x[/ilmath]s)
  • By the axiom of paring again: [ilmath]\exists B\forall b[b\in B\iff(b\in x\vee b\in \{x\})][/ilmath]
    • the [ilmath]B[/ilmath] posited to exist is written [ilmath]\{x,\{x\}\} [/ilmath]
  • By the Axiom of union: [ilmath]\exists C\forall c[c\in C\iff\exists D\in\{x,\{x\}\}(c\in D)][/ilmath]
    • We denote the [ilmath]C[/ilmath] posited to exist by [ilmath]\bigcup\{x,\{x\}\} [/ilmath] (or as a slight abuse of notation at this point: [ilmath]x\cup\{x\} [/ilmath] - as required)

See also

• Inductive set
• The natural numbers - denoted [ilmath]\mathbb{N} [/ilmath]