Frege Set Theory is Logically Inconsistent

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


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.