A Cheerful Introduction to Forcing and the Continuum Hypothesis

TL;DR

This paper provides an accessible introduction to forcing and the independence of the Continuum Hypothesis, aimed at graduate students and philosophers with minimal prior background in set theory.

Contribution

It offers a beginner-friendly explanation of forcing and its role in proving the independence of the Continuum Hypothesis from ZF axioms.

Findings

Demonstrates how forcing can be used to show independence results

Clarifies the set-theoretic method for a broad audience

Provides foundational understanding for further study in set theory

Abstract

This is an introduction to the set-theoretic method of forcing, including its application in proving the independence of the Continuum Hypothesis from the Zermelo-Fraenkel axioms of set theory. I presuppose no particular mathematical background beyond some familiarity with set theory and mathematical logic - in particular, no algebra is presupposed, though it can be useful. The goal is to have a document that makes this material accessible to mathematics graduate students in all fields, and to philosophers with an interest in set theory and mathematical logic but no other mathematical background.