The Philosophical Implications of Set Theory in Infinity

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Carson Lam Kai Shun

Abstract

What does “infinity” mean? In fact, there are mathematical, physical, and metaphysical definitions of this concept. This study will focus on the scripting of the three philosophical foundations of mathematics — formalism, intuitionism, and logic-ism — in set theory (Snapper, 1979). Various examples will be provided regarding the concept of infinity for these three schools of thought. However, none of them can prove whether there is an infinite set or the existence of infinity. As such, it forms the foundational crisis of mathematics. Further elaboration on these philosophies leads to ideas of actual, potential, and absolute boundlessness, which correspond to three basic definitions of infinity. This thesis aims to correspond these philosophies to Roger Penrose’s three world philosophy, in hope of implying the quantum mind. By employing rational proof and set theory, there is a likely possibility of building a human-like AI computer. To the authors’ best knowledge, this thesis is the first to employ set symbols which connects body, mind, and spirit. More specifically, this paper aims to become the mathematical basis for the construction of a quantum computer. Using the Basic Metaphor of Infinity, as well as cognitive mechanisms such as conceptual metaphors and aspects, one can fully appreciate the transfinite cardinals’ beauty (Nũńez, 2005). Indeed, the three mathematical philosophies map well with the three types of infinities and further fit perfectly with the body, mind, and spirit. In such a case, we may recognize how our set theory can be applied elegantly behind through mapping. This further implies the portraiture for something endless is anthropomorphic in nature or the perceptions of healing. In simple terms, because there is a connection between art and mathematics through infinity, one can enjoy the beauty of boundlessness (Maor, 1986). In essence mathematics is the science of researching the limitless.

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Research Article

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