A Renunciation of Axiom, A New Conception of Logic

Abstract

Mathematicians invented Mathematics to escape from words, but at last they depend on them just as much as everybody else. At the end, all basic definitions will be reliant on words, yet the mathematician believes that he's elevated from them by use of axioms, only that just postpones the problem and later becomes more serious G\"odelian problems due to the countability of the axiomatic array. I think we should do our best to rid ourselves of axioms, and in this paper, I revisit what's called "Na\"ive Set Theory", which is set theory that is fully reliant on verbal definitions, and resolve the problems that were once found with it in the form of paradoxes, most notably, Russell's Paradox. I believe that by this, a new approach to mathematics as a whole is presented, an approach that refers to mathematics as the science of definitions. The question "how do you define definition?" will not be a part of Mathematics but of Metamathematics and to it many of the problems known about Na\"ive Set Theory will be funneled. A thorough yet concise treatment of the matter will be given in this paper.