Search

The origin of the concepts that we employ such as natural numbers, rational and irrational numbers, zero, infinity and the place value system among others originate in the Vedas, or Veda Samhitas to be correct [Veda Samhitas]. However, due to our second-hand reception from the Arabs and the belief that everything originated in Greece, this appears to have lost the original source and meaning [V. Lakshmikantham, S. Leela, J. Vasundhara Devi, The Origin and History of Mathematics, Cambridge Scientific Publishers, Cambridge, UK, 2005. [5] ]. As a consequence, the so-called rational thinking or logic that applies to concrete situations when employed for concepts that are beyond our sensory perceptions has created some paradoxes [E.E. Escultura, FTG XVII: The new mathematics and physics, J. Appl. Math. Comput. 138 (2003) 127–149; E.E. Escultura, FTG XXXIV. Foundations of analysis and the new arithmetic, Nonlinear Anal. Phenom. III (2006) (in press)]. It is therefore necessary to return to the origin to clearly understand the basis. In this paper, we plan to propose a hybrid real number system utilizing the original ideas so as to shed some light on the existing system and strengthen it.

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].