mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premis ...

are

inconsistent In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consiste ...

, in particular the version of

Haskell Curry Haskell Brooks Curry (; September 12, 1900 – September 1, 1982) was an American mathematician and logician. Curry is best known for his work in combinatory logic. While the initial concept of combinatory logic was based on a single paper by ...

combinatory logic Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of com ...

introduced in 1930, and

Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American mathematician, computer scientist, logician, philosopher, professor and editor who made major contributions to mathematical logic and the foundations of theoretical computer scien ...

's original

lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation th ...

, introduced in 1932–1933, both originally intended as systems of formal logic. The paradox was exhibited by

Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of ...

and

J. B. Rosser John Barkley Rosser Sr. (December 6, 1907 – September 5, 1989) was an American logician, a student of Alonzo Church, and known for his part in the Church–Rosser theorem, in lambda calculus. He also developed what is now called the " Rosser si ...

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 logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully betwee ...

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.

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 {{mathlogic-stub

The paradox

See also

References