In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Kleene–Rosser paradox is a paradox that shows that certain systems of
formal logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
are
inconsistent, in particular the version of
Haskell Curry's
combinatory logic introduced in 1930, and
Alonzo Church
Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
's original
lambda calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited by
Stephen Kleene and
J. B. Rosser in 1935.
The paradox
Kleene and Rosser were able to show that both systems are able to characterize and enumerate their provably total, definable number-theoretic functions, which enabled them to construct a term that essentially replicates
Richard's paradox in formal language.
Curry later managed to identify the crucial ingredients of the calculi that allowed the construction of this paradox, and used this to construct a much simpler paradox, now known as
Curry's paradox.
See also
*
List of paradoxes
References
* Andrea Cantini,
The inconsistency of certain formal logics, in the ''Paradoxes and Contemporary Logic'' entry of ''Stanford Encyclopedia of Philosophy'' (2007).
*
Lambda calculus
Mathematical paradoxes
Self-referential paradoxes
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