Successor of a set
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- 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^{[1]}:
- [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)
- 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
See also
- Inductive set
- The natural numbers - denoted [ilmath]\mathbb{N} [/ilmath]