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\begin{document}

\title{The Independence of the Continuum Hypothesis in Isabelle/ZF}
\author{Emmanuel Gunther\thanks{Universidad Nacional de C\'ordoba. 
    Facultad de Matem\'atica, Astronom\'{\i}a,  F\'{\i}sica y
    Computaci\'on.}
  \and
  Miguel Pagano\footnotemark[1]
  \and
  Pedro S\'anchez Terraf\footnotemark[1] \thanks{Centro de Investigaci\'on y Estudios de Matem\'atica
    (CIEM-FaMAF), Conicet. C\'ordoba. Argentina.
    Supported by Secyt-UNC project 33620180100465CB.}
  \and
  Mat\'{\i}as Steinberg\footnotemark[1]
}
\maketitle

\begin{abstract}
  We redeveloped our formalization of forcing in the set theory framework of
  Isabelle/ZF. Under the assumption of the existence of a countable
  transitive model of $\ZFC$, we construct proper generic extensions 
  that satisfy the Continuum Hypothesis and its negation.
\end{abstract}


\tableofcontents

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\section{Introduction}
We formalize the theory of forcing. We work on top of the Isabelle/ZF
framework developed by \citet{DBLP:journals/jar/PaulsonG96}. Our
mechanization is described in more detail in our papers
\cite{2018arXiv180705174G} (LSFA 2018), \cite{2019arXiv190103313G},
and \cite{2020arXiv200109715G} (IJCAR 2020).

The main entry point of the present session is
\theory{Definitions\_Main.thy} (Section~\ref{sec:main-definitions}),
in which a path from fundamental set
theoretic concepts formalized in Isabelle reaching to our main theorems
is expounded. Cross-references to major milestones are provided there.

In order to provide evidence for the correctness of several of our relativized
definitions, we needed to assume the Axiom of Choice ($\AC$) during the
aforementioned theory. Nevertheless, the whole of our development is
independent of $\AC$, and the theory \theory{CH.thy} already provides
all of our results and does not import that axiom.

\subsection*{Release notes}
\label{sec:release-notes}

Previous versions of this development can be found at
\url{https://cs.famaf.unc.edu.ar/~pedro/forcing/}.

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