Editing Lambda calculus (section)

== Semantics ==
The fact that lambda calculus terms act as functions on other lambda calculus terms, and even on themselves, led to questions about the semantics of the lambda calculus. Could a sensible meaning be assigned to lambda calculus terms? The natural semantics was to find a set ''D'' isomorphic to the function space ''D'' → ''D'', of functions on itself. However, no nontrivial such ''D'' can exist, by [[cardinality]] constraints because the set of all functions from ''D'' to ''D'' has greater cardinality than ''D'', unless ''D'' is a [[singleton set]].

In the 1970s, [[Dana Scott]] showed that if only [[Scott continuity|continuous functions]] were considered, a set or [[Domain theory|domain]] ''D'' with the required property could be found, thus providing a [[Model theory|model]] for the lambda calculus.<ref>{{cite journal
| last1      = Scott
| first1     = Dana
| date       = 1993
| title      = A type-theoretical alternative to ISWIM, CUCH, OWHY
| url        = https://www.cs.cmu.edu/~crary/819-f09/Scott93.pdf
| journal    = Theoretical Computer Science
| volume     = 121
| issue = 1–2
| pages      = 411–440
| doi = 10.1016/0304-3975(93)90095-B
| access-date = 2022-12-01
}} Written 1969, widely circulated as an unpublished manuscript.</ref>

This work also formed the basis for the [[denotational semantics]] of programming languages.