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Explore the mathematical foundations of combinatory logic through a colloquium presentation that examines combinatory completeness within structured multicategories. Delve into the century-old pursuit of logical syntax without bound variables and discover how combinatory algebras serve as algebraic models of combinatory logic, consisting of sets with binary operations satisfying complex combinatory completeness conditions. Learn about the generalization of combinatory completeness relative to faithful Cartesian clubs - well-behaved collections of functions between finite sets - where classical combinatory completeness represents the club of all such functions. Understand how the equivalence between combinatory completeness and the existence of elements satisfying simple equations extends to various instances of this generalized notion. Examine how these mathematical results apply beyond the traditional category of sets and functions, with technical development occurring within multicategories equipped with faithful Cartesian club actions. Gain insights into structured multicategories as natural frameworks for studying variations of combinatory algebras and their applications in logic and computer science.
