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\begin{center}{\huge Closed Sets and Class Forcing}\end{center}
\begin{center}{\large Mack Stanley}\end{center}
\begin{center}{\large San Jose}\end{center}

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\large

Assume the~GCH\null. If $S$ is a set of ordinals, say that $C$ is a {\it club
subset\/} of $S$ when $C$ is closed and $\sup(C)=\sup(S)$.  It is well known
that if $S$ is an unbounded subset of~$\aleph_{1}$, then $S$ has a club subset
in an $\aleph_{1}$-preserving outer model iff $S$ is stationary
in~$\aleph_{1}$ iff $S$~has a club subset in a cardinal preserving set generic
extension. If $S$ is an unbounded subset of~$\aleph_{2}$, then $S$ may have a
club subset in an $\aleph_{2}$-preserving class generic extension, but none in
any $\aleph_{2}$-preserving set generic extension.
\par
Suppose that $\kappa\ge\aleph_{\alpha+1}$ is regular and that $S$ is unbounded
in $\kappa$ and has a club subset in an outer model in which
$\kappa\ge\aleph_{\alpha+1}$.  Depending on $\alpha$ and the ``pattern width''
of~$S$, class forcing may be required to produce an outer model in which $S$
has a club subset and $\kappa\ge\aleph_{\alpha+1}$.}

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