In
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 premises ...
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 consistent ...
, 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 ...
's
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 comput ...
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 scienc ...
'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 ...
, 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 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
Curry's paradox is a paradox in which an arbitrary claim ''F'' is proved from the mere existence of a sentence ''C'' that says of itself "If ''C'', then ''F''", requiring only a few apparently innocuous logical deduction rules. Since ''F'' is arbi ...
.
See also
*
List of paradoxes
This list includes well known paradoxes, grouped thematically. The grouping is approximate, as paradoxes may fit into more than one category. This list collects only scenarios that have been called a paradox by at least one source and have their ...
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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