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\documentclass{article}
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\lecture{22}                  %% Lecture number
\title{Scott's $D_\infty$ Construction}   %% Title of lecture
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As observed in Lecture 19, one of the problems with modeling the untyped $\lambda$-calculus is that we cannot have a nontrivial domain $D$ isomorphic to its function space $D\to D$ because of cardinality restrictions.  However, Dana Scott showed that given any pointed CPO $D$, it is possible to embed $D$ into a pointed CPO $D_\infty$ that is isomorphic to its \emph{continuous} function space $[D_\infty\to D_\infty]$.  This construction allows us to give a denotational model of the untyped $\lambda$-calculus.

The notes on the following pages give an overview of Scott's construction.  The notes are extracted from

\medskip\noindent
Leonid Rudin, $\lambda$-Logic.  Technical Report 4521, Computer Science Department, California Institute of Technology, May 1981. 


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