§ 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; .
- This is impossible for cardinality constraints.
- Key idea: restrict to continuous functions! .
- Solutions exist! Eg. space of continuous with appropriate topology is like space of "eventually stabilizing sequences", which is equinumerous to , since sequences that eventually become stable have information . This has the same cardinality as .
- For continuity in general, we need a topology .
- OK, now that we know this is what we need, how do we exhibit a space ? One invokes the hammer of domain theory
- Now that we have the space , what's the right topology on it? That's worth a turing award! The Scott topology