Independence of the Continuum Hypothesis: an Intuitive Introduction

TL;DR

This paper provides an accessible introduction to the independence of the continuum hypothesis, explaining the key concepts and techniques like forcing to a broad audience of non-specialists.

Contribution

It offers an intuitive, comprehensive exposition of the independence proof of the continuum hypothesis aimed at advanced undergraduates and computer scientists.

Findings

Clarifies the role of forcing in set theory

Demonstrates the independence of the continuum hypothesis

Bridges the gap between set theory and broader mathematical education

Abstract

The independence of the continuum hypothesis is a result of broad impact: it settles a basic question regarding the nature of N and R, two of the most familiar mathematical structures; it introduces the method of forcing that has become the main workhorse of set theory; and it has broad implications on mathematical foundations and on the role of syntax versus semantics. Despite its broad impact, it is not broadly taught. A main reason is the lack of accessible expositions for nonspecialists, because the mathematical structures and techniques employed in the proof are unfamiliar outside of set theory. This manuscript aims to take a step in addressing this gap by providing an exposition at a level accessible to advanced undergraduate mathematicians and theoretical computer scientists, while covering all the technically challenging parts of the proof.