The set-theoretic antinomies certainly brought to light, as the above proof shows, the classical inconsistency of naive set theory, indeed its inconsistency in logics considerably weaker than classical logic--intuitionistic logic, relevant logics at least as strong as (the rather weak) T or
RWX etc.(5) The almost universal response has been to abandon it as false, a salutary reminder of the fallibility of mathematical intuition.(6) But there have been other responses, many motivated by dissatisfaction with the hierarchical view of set theory which underlies the standard reply.