The hybrid set theory

We shall develop, in this paper, the hybrid set theory extending some notions of the hybrid real number system considered in [V. Lakshmikantham, D. Kovach, Hybrid real number system, Nonlinear Analysis: Hybrid Systems (2006), doi:10.1016/nahs.2006.07.001], so as to shed some light on the existing set theory and strengthen it. We shall also list some ideas of set theory that existed long before Cantor’s discovery [G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers (P.E.B., Trans.), The Open Court Publishing Co., la Salle, Illinois, 1952].