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 comprehension principle 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.
- 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