Singleton (set theory)

Definition

Let $X$ be a set. We call $X$ a singleton if:

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

More concisely this may be written:

• $\exists t\in X\forall s\in X[t\eq s]$
  • 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.

See next

• A pair of identical elements is a singleton

Proof of claims

1. $\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X\forall s\in X[t\eq s]\big)$
   • By "rewriting for-all and exists within set theory" we see:
     • $\big(\exists t[t\in X\wedge\forall s(s\in X\rightarrow s\eq t)]\big)\iff\big(\exists t\in X[\forall s(s\in X\rightarrow s\eq t)]\big)$

       $\iff\big(\exists t\in X\forall s(s\in X\rightarrow s\eq t)\big)$ by simplification

       $\iff\big(\exists t\in X\forall s[s\in X\rightarrow s\eq t]\big)$ by changing the bracket style

       $\iff\big(\exists t\in X\forall s\in X[s\eq t]\big)$ by re-writing again.