The continuum hypothesis: Its independence from Zermelo-Fraenkel set theory and impact on mathematical foundations

Abstract

The Continuum Hypothesis, originally posited by the pioneering mathematician Georg Cantor in the latter part of the 19th century, stands as a cornerstone inquiry in the realm of set theory. This paper embarks on a journey, delving into the rudiments of set theory, before tracing the evolutionary trajectory of the Continuum Hypothesis. Central to this exploration is the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) a foundational pillar in modern set theoretic studies. The core tenets of ZFC are dissected, shedding light on the seminal proofs presented by luminaries in the field that underline the unprovability of the CH within this axiomatic system. Beyond its mathematical intricacies, the paper underscores the profound philosophical and practical implications of the CH in both set theory and the broader mathematical landscape. In synthesizing these insights, a profound realization emerges: the inherent limitations in establishing the veracity of the Continuum Hypothesis within the confines of ZFC. This poignant revelation beckons deeper introspection into the foundational underpinnings of mathematics, stirring both intrigue and reflection amongst scholars and enthusiasts alike.