All About the Meaning of E, a Symbol of Being an Element of a Set, as Being Part of that Which Has No Part

Abstract

In the study of Set Theory, few have ever brought forth the question on the meaning of ϵ, the symbol denoting the most central relation between two objects in the domain. This is similar to the study of Geometry in which few have ever questioned the meaning of the most central notion Point. It is also similar to the study of Phenomenology in which few have ever raised the question on the meaning of the most central concept Being until Heidegger in 1927. In this exchange note following “Between ‘A Brief History of Analytic Philosophy: from Russell to Rawls’ and ‘Logico-Philosophicus Serial Booklet’” (abbrev as ‘Between …’ note), we bring forth the question on the meaning of ϵ and make a trying effort to find the answer to it. We first show the difference and the thing in common between Russell's own resolution to Russell’s Paradox and the one proposed by us in the ending part of ‘Between …’ note to directly address Professor Schwartz’s critical comment on the seeming similarity between these two. Then, with the help of Frege’s Concept of Functions, Russell’s Naive Set and Zermelo’s Definition of ‘Definite’, we demonstrate in order that (1) the two resolutions to Russell’s Paradox are both equal and necessary to each other; (2) the Russell’s Paradox, the Naive Set in particular, essentially reveals the lack of our understanding on ϵ being argument of a propositional function; (3) ϵ, the mathematical symbol denoting being an element of [a set into a whole] according to the words of Russia-born Georg Cantor, is meant for being part of [that which has no part] in the language of Euclid of Alexandria. In the end, we extend the discussion to show that the final outcome of our analysis in this article are also tightly related with (1) the Dasein concept investigated by Heidegger (2) the Point & Line notion explored by Euclid (3) Continuum Hypothesis initiated by Georg Cantor and (4) ‘Logico-Philosophicus Serial Booklet’, an article series authored by us that revisit the central issues in Philosophy of Language and Logic (Meaning of a Name), Philosophy of Phenomenology (Being of a Thing) and Classic Axioms in Science (Exact Meaning and Logic of Definitions and Postulates) respectively, with the hope of shedding light on a few critical concepts in the study of Science and Philosophy.