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:

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

More concisely this may be written:

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

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