# Singleton (set theory)

## Definition

Let [ilmath]X[/ilmath] be a set. We call [ilmath]X[/ilmath] a singleton if:

• [ilmath]\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)][/ilmath]Caveat:See:[Note 1]
• In words: [ilmath]X[/ilmath] is a singleton if: there exists a thing such that ( the thing is in [ilmath]X[/ilmath] and for any stuff ( if that stuff is in [ilmath]X[/ilmath] then the stuff is the thing ) )

More concisely this may be written:

• [ilmath]\exists t\in X\forall s\in X[t\eq s][/ilmath][Note 2]
• For proof see Claim 1.

## Significance

Notice that we have manage to define a set containing one thing without any notion of the number 1.

## Proof of claims

1. [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X\forall s\in X[t\eq s]\big)[/ilmath]
• By "rewriting for-all and exists within set theory" we see:
• [ilmath]\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)[/ilmath]
[ilmath]\iff\big(\exists t\in X\forall s(s\in X\rightarrow s\eq t)\big)[/ilmath] by simplification
[ilmath]\iff\big(\exists t\in X\forall s[s\in X\rightarrow s\eq t]\big)[/ilmath] by changing the bracket style
[ilmath]\iff\big(\exists t\in X\forall s\in X[s\eq t]\big)[/ilmath] by re-writing again.