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\noindent{\LARGE \bf \sf Iterated forcing and the Continuum Hypothesis}

\vskip.25in

In June 1965 --- barely two years after Cohen's discovery of forcing --- Robert Solovay and Stanley Tennenbaum introduced the technique of iterated forcing in their proof of the consistency of Souslin's Hypothesis~\cite{MR0294139}.  In the four decades following, iterated forcing has assumed a central role in establishing consistency results.  The associated technology has grown more and more sophisticated over the years, and many times advances in set theory have been fueled by corresponding breakthroughs in iteration theory. Although we know much about iterated forcing, there are still many general questions, and in this workshop we address one such:

\medskip
\begin{center}
{\em Why is it so difficult to use iterated forcing to produce ``interesting'' models of {\sf ZFC + CH}?}
\end{center}

\medskip

 What sorts of difficulties arise? Consider, for example, the problem of obtaining a model of {\sf CH} in which Souslin's Hypothesis ({\sf SH}) holds. Given a Souslin tree, it is well-known that one can generically add a branch through the tree (killing its Souslinity) while not adding new reals to the ground model: one transforms the tree into a notion of forcing by ``turning it upside-down''.  The natural way to proceed would be to start with a model of {\sf GCH} and then do an iteration of the above forcings, making sure that any potential Souslin tree is killed off at some stage of the iteration. This procedure is sufficient for obtaining a model of {\sf SH}, and one might hope, because we are not adding reals at successor stages of the iteration, that the Continuum Hypothesis would be preserved by the iteration.  Jensen~(see~\cite{MR0384542}) shows that this naive approach runs into problems --- even though forcing with an individual Souslin tree does not add new reals to the ground model, iterations of such forcings very well might. A large part of the monograph~\cite{MR0384542} is devoted to an exposition of how Jensen managed to overcome these issues and produce a model of {\sf ZFC + CH + SH}.

 Things have progressed significantly since Jensen's work in the late 1960s.  For example, a close study of the way in which forcing with Souslin trees can add new reals culminated in the isolation of {\em weak diamond} by Devlin and Shelah~\cite{MR0469756}. In 1984, Shelah's seminal monograph {\em Proper Forcing}~\cite{MR675955} introduced the notion of $\mathbb{D}$-completeness --- a property  that is a key ingredient proofs that iterations do not add new reals. The twenty-five years since {\em Proper Forcing} have seen our toolkit for ensuring that the Continuum Hypothesis is preserved in iterated forcing arguments grow substantially, but at a cost --- even though many important results have been obtained, the methods used have grown so complex that they are understood by only a handful of experts.  This workshop is intended to remedy the situation.

 Our workshop will be organized around two guiding principles. First, we want to give attention to examples of ``single-step'' forcings, that is, partial orders that accomplish a particular task without adding reals.  Second, we wish to give an account of how iterations can be done without adding new reals. Many examples will be presented, and many open questions will be formulated --- the talks will be pedagogical, and not merely the reporting of results. Our intent is that participants should leave with a firm understanding of how the basic iteration theorems work, in addition to acquiring a reasonable knowledge of the current state of affairs.


\begin{thebibliography}{1}

\bibitem{MR0384542}
Keith~J. Devlin and H{\.a}vard Johnsbr{\.a}ten.
\newblock {\em The {S}ouslin problem}.
\newblock Lecture Notes in Mathematics, Vol. 405. Springer-Verlag, Berlin,
  1974.

\bibitem{MR0469756}
Keith~J. Devlin and Saharon Shelah.
\newblock A weak version of {$\diamondsuit $} which follows from {$2\sp{\aleph
  \sb{0}}<2\sp{\aleph \sb{1}}$}.
\newblock {\em Israel J. Math.}, 29(2-3):239--247, 1978.

\bibitem{MR675955}
Saharon Shelah.
\newblock {\em Proper forcing}, volume 940 of {\em Lecture Notes in
  Mathematics}.
\newblock Springer-Verlag, Berlin, 1982.

\bibitem{MR0294139}
R.~M. Solovay and S.~Tennenbaum.
\newblock Iterated {C}ohen extensions and {S}ouslin's problem.
\newblock {\em Ann. of Math. (2)}, 94:201--245, 1971.

\end{thebibliography}


\end{document}
