§ Denotational semantics in a few sentences

• We want to find a math object that reflects lambda calculus
• Such an object must contain its own space of functions; $L \simeq [L \to L]$.
• This is impossible for cardinality constraints.
• Key idea: restrict to continuous functions! $L \simeq [L \xrightarrow{\texttt{cont}} L]$.
• Solutions exist! Eg. space of continuous $[\mathbb N \to \mathbb N]$ with appropriate topology is like space of "eventually stabilizing sequences", which is equinumerous to $\mathbb N$, since sequences that eventually become stable have information $\cup_{i=0}^\infty \mathbb N^i$. This has the same cardinality as $\mathbb N$.
• For continuity in general, we need a topology .
• OK, now that we know this is what we need, how do we exhibit a space $L \simeq [L \to L]$? One invokes the hammer of domain theory
• Now that we have the space $L$, what's the right topology on it? That's worth a turing award! The Scott topology