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 Axiom of Abstraction 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.

$\blacksquare$

Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 8$ Russell's paradox