The logic of sheaves, sheaf forcing and the independence of the Continuum Hypothesis

TL;DR

This paper introduces a logical framework using sheaves of structures and sheaf forcing to provide a simplified, unified proof of the independence of the Continuum Hypothesis, avoiding complex categorical machinery.

Contribution

It offers an alternative proof of the Continuum Hypothesis independence using sheaf logic, simplifying and unifying previous boolean and intuitionistic methods.

Findings

Simplifies the proof of the Continuum Hypothesis independence

Unifies boolean and intuitionistic approaches

Avoids categorical complexities of topoi

Abstract

An introduction is given to the logic of sheaves of structures and to set theoretic forcing constructions based on this logic. Using these tools, it is presented an alternative proof of the independence of the Continuum Hypothesis; which simplifies and unifies the classical boolean and intuitionistic approaches, avoiding the difficulties linked to the categorical machinery of the topoi based approach.