Empty Set can be Derived from Axiom of Abstraction

Theorem

The empty set can be formed by application of the Axiom of Abstraction.

Hence the empty set can be derived as a valid object in Frege set theory.

Let $P$ be the property defined as:

$\forall x: \map P x := \neg \paren {x = x}$

Hence, using the Axiom of Abstraction, we form the set:

$\O := \set {x: \neg \paren {x = x} }$

where the property ${\map P x}$ is:

$\neg \paren {x = x}$

Since we have that:

$\forall x: x = x$

it is seen that $\O$ as defined here has no elements.

By definition of Frege set theory, given any property $P$, there exists a unique set which consists of all and only those objects which have property $P$:

$\set {x: \map P x}$

Therefore, there exists a unique set

$\O := \set {x: \neg \paren {x = x} }$

which the property $P= \neg \paren {x = x}$, which consists of no elements.

Hence, by the definition of the empty set, this set is a valid object in Frege set theory.

Hence the result by definition of the empty set.

$\blacksquare$