Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)This article intends to contribute to the debate about the uses of paraconsistent reasoning in the foundations of set theory, by means of using the logics of formal inconsistency and by considering consistent and inconsistent sentences, as well as consistent and inconsistent sets. We establish the basis for new paraconsistent set-theories (such as ZFmbC and ZFCil) under this perspective and establish their non-triviality, provided that ZF is consistent. By recalling how George Cantor himself, in his efforts towards founding set theory more than a century ago, not only used a form of 'inconsistent sets' in his mathematical reasoning, but regarded contradictions as beneficial, we argue that Cantor's handling of inconsistent collections can be related to ours.26 97116FAPESP (Thematic Project LogCons, Brazil) [2010/51038-0]National Council for Scientific and Technological Development (CNPq), BrazilFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)