Formal Presentation
Define Naive Set Theory (NST) as the theory of predicate logic with a binary predicate and the following axiom schema of unrestricted comprehension:
for any formula P with only the variable x free. Substitute for . Then by existential instantiation (reusing the symbol y) and universal instantiation we have
a contradiction. Therefore NST is inconsistent.
Read more about this topic: Russell's Paradox
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