The inconsistency of a certain axiom system for set theory

We prove in this paper that the -system, an axiom system for set theory suggested for investigation by Takeuti in [2], is inconsistent. We also show that this system without the ω-rule is consistent if Zermelo-Fraenkel set theory with the axiom of choice and an axiom due to Reinhardt and Silver is consistent. The -system is an effort to strengthen Bernays-Gödel set theory by adding a reflection principle. In addition to the standard notation of set theory, we write X″{x) to mean {y∣〈x, y〉 Є X}.