In this paper we study the systemsP andP * (see Arruda and da Costa,O paradoxo de Curry-Moh Shaw-Kwei, Boletim da Sociedade Matemtica de São Paulo 18 (1966)) and some related systems. In the last section, we prove that certain set theories havingP andP * as their underlying logics are non-trivial.
(or not oveA-complete.) . Let * be a unary operator defined on the set F of formulas of the language £ (ie, if A is a formula of £, then *A is also a ...
