Singleton Axiom (First-Order Logic)

Definition

Singleton Axiom (First-Order Logic)

In the language of set theory with the membership relation symbol , the singleton axiom is the sentence

For every , there is a set whose only member is : the singleton .

Entailment of Weak Singleton Axiom

Theorem

The singleton axiom entails the weak singleton axiom:

Proof

Let be a model of the singleton axiom and choose any in its domain. The axiom gives a such that, for every ,

Taking gives . Thus every belongs to some , so satisfies the weak singleton axiom.