The consistency of the negation of the Continuum Hypothesis

Abstract

[en] The purpose of this work is to prove the consistency of the negation of the Continuum Hypothesis $(\mathrm{CH})$ with the Zermelo - Fraenkel axiomatic system, including the Axiom of Choice (ZFC). The Continuum Hypothesis states that there is no set whose cardinality is strictly between the cardinality of the set of integers and the cardinality of the set of real numbers. It is well-known that $C H$ is independent of ZFC: neither $C H$ nor its negation can be proved from ZFC. In order to show the consistency of $\neg C H$, we will use the method of forcing that permits us to construct a model that satisfies all the axioms of $Z F C$ and where $C H$ fails.