Frege Set Theory is Logically Inconsistent

Theorem
The system of axiomatic set theory that is Frege set theory is inconsistent.

Proof
From Russell's Paradox, the leads to a contradiction.

Let $q$ be such a contradiction:
 * $q = p \land \neg p$

for some statement $p$.

From the Rule of Explosion it then follows that every logical formula is a provable consequence of $q$.

Hence the result, by definition of inconsistent.