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Theorem in order theory and lattice theory
theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem. Suppose
Kleene_fixed-point_theorem
Condition for a mathematical function to map some value to itself
space Kakutani fixed-point theorem Kleene fixed-point theorem Knaster–Tarski theorem Lefschetz fixed-point theorem Nielsen fixed-point theorem Poincaré–Birkhoff
Fixed-point_theorem
Theorem in computability theory
proved by Stephen Kleene in 1938 and appear in his 1952 book Introduction to Metamathematics. A related theorem, which constructs fixed points of a computable
Kleene's_recursion_theorem
American mathematician (1909–1994)
hierarchy, Kleene algebra, the Kleene star (Kleene closure), Kleene's recursion theorem and the Kleene fixed-point theorem. He also invented regular expressions
Stephen_Cole_Kleene
Theorem in order and lattice theory
of L, thus giving a more "constructive" version of the theorem. (See: Kleene fixed-point theorem.) More generally, if f is monotonic, then the least fixpoint
Knaster–Tarski_theorem
Smallest fixed point of a function from a poset
restrictions (see Kleene fixed-point theorem), which are met in the example, F {\displaystyle F} necessarily has a least fixed point, fact {\displaystyle
Least_fixed_point
Topics referred to by the same term
theorem (sometimes referred to as Tarski's fixed point theorem) Tarski–Seidenberg theorem Some fixed point theorems, usually variants of the Kleene fixed-point
Tarski's_theorem
Tarski's undefinability theorem Tarski–Seidenberg theorem Some fixed point theorems, usually variants of the Kleene fixed-point theorem, are referred to the
List of things named after Alfred Tarski
List_of_things_named_after_Alfred_Tarski
embedding theorem (ordered groups) Hausdorff maximality theorem (set theory) Kleene fixed-point theorem (order theory) Knaster–Tarski theorem (order theory)
List_of_theorems
Limitative results in mathematical logic
about undecidable sets in recursion theory. Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability theory
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Fixed-point theorem
mathematics, the Bourbaki–Witt theorem in order theory, named after Nicolas Bourbaki and Ernst Witt, is a basic fixed-point theorem for partially ordered sets
Bourbaki–Witt_theorem
Topics referred to by the same term
Recursion theorem can refer to: The recursion theorem in set theory Kleene's recursion theorem, also called the fixed point theorem, in computability
Recursion_theorem
Mathematical phrase
f n(⊥), ...) of ⊥ (see also the Kleene fixed-point theorem). Another fixed point theorem is the Bourbaki–Witt theorem, stating that if f {\displaystyle
Complete_partial_order
Self-replicating program
JavaScript Machine, with a series of interactive hints A Java Quine built straight from Kleene's fixed point theorem, composition and s-n-m A QR code quine
Quine_(computing)
Problem in computer science
1965, p. 115 Lucas 2021. Kleene 1952, p. 382. Rosser, "Informal Exposition of Proofs of Gödel's Theorem and Church's Theorem", reprinted in Davis 1965
Halting_problem
Mathematical logic concept
in 1982 that Goodstein's theorem cannot be proven in Peano arithmetic. Their proof was based on Gentzen's theorem. See Kleene (2009, pp. 476–499) for a
Gentzen's_consistency_proof
Subfield of automated reasoning and mathematical logic
2026-01-25. Kleene, Stephen Cole (1967). Mathematical Logic. Mineola, N.Y.: Dover Publications. Raatikainen, Panu (2026), "Gödel's Incompleteness Theorems", in
Automated_theorem_proving
Formal semantics of logic programming languages
on T. By the Knaster–Tarski theorem, this map has a least fixed point; by the Kleene fixed-point theorem the fixed point is the supremum of the chain
Syntax and semantics of logic programming
Syntax_and_semantics_of_logic_programming
Topics referred to by the same term
first incompleteness theorem Tarski's undefinability theorem Halting problem Kleene's recursion theorem Lawvere's fixed-point theorem (categorical generalization
Diagonal_argument
Theorem about fixed points of multiple variables
the product order (componentwise order). By the Kleene fixed-point theorem, it has a least fixed point μ ( x , y ) . ( f , g ) ( x , y ) {\displaystyle
Bekić's_theorem
Impossible task in computing
they are equivalent or not. He relied heavily on earlier work by Stephen Kleene. Turing reduced the question of the existence of an 'algorithm' or 'general
Entscheidungsproblem
Logical quantifier
edu. Retrieved 2019-12-14. This is a consequence of the compactness theorem. Kleene, Stephen (1952). Introduction to Metamathematics. Ishi Press International
Uniqueness_quantification
Mathematical-logic system
shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox. Subsequently, in 1936 Church isolated
Lambda_calculus
Mathematical logic concept
Kleene, Stephen Cole (1967). Mathematical Logic. Wiley. ISBN 9780471490333. Klenk, Virginia (1976). "Intended Models and the Löwenheim-Skolem Theorem"
Skolem's_paradox
System including an indeterminate value
or false, but in many cases we don't know which. Similarly, Stephen Cole Kleene used a third value to represent predicates that are "undecidable by [any]
Three-valued_logic
Generalised alphabetical order
greater monomial) is a multiple of this least indeterminate. Collation Kleene–Brouwer order Lexicographic preferences – an application of lexicographic
Lexicographic_order
Type of logical system
subformula It seems that symbol ⊨ {\displaystyle \vDash } was introduced by Kleene; see footnote 30 in Dover's 2002 reprint of his book Mathematical Logic
First-order_logic
Dutch mathematician and logician
Brouwer proved a number of theorems in the emerging field of topology. The most important were his fixed point theorem, the topological invariance of
L._E._J._Brouwer
Generalization of Rice's theorem
p {\displaystyle p} can get access to its own source code by Kleene's recursion theorem). If this eventually returns true, then this first task continues
Rice–Shapiro_theorem
Logical principle
since he does not conceive the natural numbers as a completed totality. (Kleene 1952:49–50) Hilbert and Brouwer both give examples of the law of excluded
Law_of_excluded_middle
Non-contradiction of a theory
and this formula is said to be (formally) provable or be a (formal) theorem" cf Kleene 1952, p. 83. Carnielli, Walter; Coniglio, Marcelo Esteban (2016).
Consistency
Size of a set in mathematics
116, 118 Enderton 1977, pp. 128–129 Kleene 1952, p. 3 Suppes 1972, p. 91 Tao 2022, pp. 57–58 Halmos 1998, p. 52 Kleene 1952, p. 9 Kuratowski 1968, p. 169
Cardinality
Branch of mathematics relating to posets
(f)=\bigsqcup _{n\in \mathbb {N} }f^{n}(\bot ).} This is the Kleene fixed-point theorem. The ⊔ {\displaystyle \sqcup } symbol is the directed join. A
Domain_theory
Study of computable functions and Turing degrees
the work of Kurt Gödel, Alonzo Church, Rózsa Péter, Alan Turing, Stephen Kleene, and Emil Post. The fundamental results the researchers obtained established
Computability_theory
Computation model defining an abstract machine
the left of the scanned symbol. A variant of this is seen in Kleene (1952) where Kleene shows how to write the Gödel number of a machine's "situation":
Turing_machine
Concept in the philosophy of mathematics
University Press. p. 271. ISBN 9780838631393. OCLC 230508222. Kleene 1952/1971:48. Kleene 1952/1971:48 p. 357; also "the machine ... is supplied with a
Actual_and_potential_infinity
Thesis on the nature of computability
i.e. by one of his machines, is equivalent to Church's thesis by Theorem XXX. Kleene, finally, uses for the first time the term the "Church-Turing thesis"
Church–Turing_thesis
Subfield of mathematics
compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are fixed-point logics that
Mathematical_logic
Sequence of characters that forms a search pattern
2020-10-07. Retrieved 2017-12-10. Kozen, Dexter (1991). "A completeness theorem for Kleene algebras and the algebra of regular events". [1991] Proceedings Sixth
Regular_expression
Foundational controversy in twentieth-century mathematics
he had published a number of important papers, in particular the fixed-point theorem. Hilbert admired Brouwer and helped him receive a regular academic
Brouwer–Hilbert_controversy
Basic framework of mathematics
generating self-contradictory theories, and to have reliable concepts of theorems, proofs, algorithms, etc. in particular. This may also include the philosophical
Foundations_of_mathematics
In logic, a statement which is always true
from a given tautology (Kleene 1967 sec. 3). Suppose that S is a tautology and for each propositional variable A in S a fixed sentence SA is chosen. Then
Tautology_(logic)
Statement in mathematical logic
lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence of self-referential sentences in certain
Diagonal_lemma
Syntactically correct logical formula
all introductory textbooks, including Enderton (2001), Gamut (1990), and Kleene (1967) Gensler, Harry (2002-09-11). Introduction to Logic. Routledge. p
Well-formed_formula
Sequence of words formed by specific rules
set of all words over an alphabet Σ is usually denoted by Σ* (using the Kleene star). The length of a word is the number of letters it is composed of.
Formal_language
Subfield of set theory
ordering of the ordinals agrees with the Kleene–Brouwer order on T s {\displaystyle T_{s}} . Recall that Kleene–Brouwer order is like lexicographical order
Determinacy
can be disproven from the standard axioms of set theory. 1943 - Stephen Kleene introduces the assertion he calls "Church's Thesis" asserting the identity
Timeline of mathematical logic
Timeline_of_mathematical_logic
Method of deriving conclusions
between. In many-valued logics, some propositions are neither true nor false. Kleene logic, for example, is a three-valued logic that introduces the additional
Rule_of_inference
3-volume treatise on mathematics, 1910–1913
taken from Kleene 1952, p. 69 substituting → for ⊃. Kleene 1952, p. 71, Enderton 2001, p. 15. Enderton 2001, p. 16. This is the word used by Kleene 1952, p
Principia_Mathematica
Ordinals in mathematics and set theory
a theorem of Friedman, Jensen, and Sacks, the countable admissible ordinals are exactly those constructed in a manner similar to the Church–Kleene ordinal
Large_countable_ordinal
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
Kind of proof calculus
of each line of proof to indicate dependencies. This is equivalent to Kleene's vertical bars. (It is not totally clear if Quine's asterisk notation appeared
Natural_deduction
Generalization of "n-th" to infinite cases
ordinal that limits a system of construction in this manner is the Church–Kleene ordinal, ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} (despite the
Ordinal_number
Theory of truth in the philosophy of language
of Tarski's logic of totally defined truth predicates) with the strong Kleene evaluation scheme. Disquotational principle Semantics of logic T-schema
Semantic_theory_of_truth
quasiregular if μ a {\displaystyle \mu _{a}} has a fixed point, which need not be unique. Each such fixed point is called a left quasi-inverse of a. If b is
Quasiregular_element
Paradox in set theory
Curry), which does not require negation Girard's paradox in type theory The Kleene–Rosser paradox, showing that the original lambda calculus is inconsistent
Russell's_paradox
System of formal deduction in logic
485–489) and Luitzen Egbertus Jan Brouwer's (1927) response (pp. 490–495) Kleene, Stephen Cole (1952). Introduction to Metamathematics (10th impression with
Hilbert_system
Mathematical set of all subsets of a set
existential quantifier is the left adjoint. Cantor's theorem Family of sets Field of sets Combination Kleene star The notation 2S, meaning the set of all functions
Power_set
Function computable with bounded loops
all primitive recursive. The following examples and definitions are from Kleene 1974, pp. 222–231. Many appear with proofs. Most also appear with similar
Primitive_recursive_function
Computer science and recursion theory
minimization operator. . .. The McCarthy formalism is like the general recursive (Kleene) system, in being based on some basic functions, composition, and equality
McCarthy_Formalism
Mathematical theory of data types
lambda calculus. Church's theory of types helped the formal system avoid the Kleene–Rosser paradox that afflicted the original untyped lambda calculus. Church
Type_theory
Whether a decision problem has an effective method to derive the answer
are not adequately represented by the set of theorems alone. (For example, Kleene's logic has no theorems at all.) In such cases, alternative definitions
Decidability_(logic)
Attempts to formalize the concept of algorithms
appears as his Theorem XXVIII. Together these form the proof of their equivalence, Kleene's Theorem XXX. With his Theorem XXX Kleene proves the equivalence
Algorithm_characterizations
Church's thesis by Theorem XXX." Indeed immediately before this statement, Kleene states the Theorem XXX: "Theorem XXX (= Theorems XXVIII + XXIX). The
History of the Church–Turing thesis
History_of_the_Church–Turing_thesis
Function that is its own inverse
odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem. The graph of an involution (on the real
Involution_(mathematics)
other operations. The set of all binary strings is denoted by {0,1}*, using Kleene star. Arbitrary subsets of {0,1}* are sometimes identified with trees, specifically
S2S_(mathematics)
incompleteness theorem. After Post completed his version of incompleteness he then added the following: "The conclusion is unescapable that even for such a fixed, well
Creative_and_productive_sets
expressed using generalized regular expressions with limited nesting depths of Kleene stars? For which number fields does Hilbert's tenth problem hold? Kueker's
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Basic notion of sameness in mathematics
Metaphysics Research Lab, Stanford University. Retrieved 20 January 2025. Kleene 1967, pp. 158–161. Suppes, Patrick (1957). Introduction to Logic (PDF).
Equality_(mathematics)
Mathematical model for deduction or proof systems
formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms. In 1921, David Hilbert proposed to use formal systems as the
Formal_system
Size of a possibly infinite set
Eric W. "Cardinal Number". mathworld.wolfram.com. Retrieved 2020-09-06. Kleene 1952, p. 9 Enderton 1977, p. 136 Pinter 2014, Page 2 of Chapter 8 Potter
Cardinal_number
Relationship where one statement follows from another
introduced by Frege in 1879, but its current use only dates back to Rosser and Kleene (1934–1935). Syntactic consequence does not depend on any interpretation
Logical_consequence
Mathematical paradox
Löb's paradox after Martin Hugo Löb, due to its relationship to Löb's theorem. Claims of the form "if A, then B" are called conditional claims. Curry's
Curry's_paradox
Base set of symbols with which a language is formed
their length) is indicated by the Kleene star operator as Σ ∗ {\displaystyle \Sigma ^{*}} , and is also called the Kleene closure of Σ {\displaystyle \Sigma
Alphabet_(formal_languages)
Class of problems solvable in polynomial time
in P are also closed under reversal, intersection, union, concatenation, Kleene closure, inverse homomorphism, and complementation. Some problems are known
P_(complexity)
Structure of a formal language
N)^{*}\rightarrow (\Sigma \cup N)^{*}} where ∗ {\displaystyle {*}} is the Kleene star operator and ∪ {\displaystyle \cup } denotes set union. That is, each
Formal_grammar
Mathematical technique used in proof theory
proof-theoretic ordinal of any theory is less than or equal to the Church–Kleene ordinal ω 1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} . In particular
Ordinal_analysis
Surname list
Brouwer fixed-point theorem, Brouwer–Heyting–Kolmogorov interpretation, Brouwer–Hilbert controversy, Kleene–Brouwer order, Phragmen–Brouwer theorem Leo Brouwer
Brouwer
Proof by Alan Turing
to the Entscheidungsproblem". It was the second proof (after Church's theorem) of the negation of Hilbert's Entscheidungsproblem; that is, the conjecture
Turing's_proof
Complexity class used to classify decision problems
Turing machines. NP is closed under union, intersection, concatenation, Kleene star and reversal. It is not known whether NP is closed under complement
NP_(complexity)
Axiomatic set theories based on the principles of mathematical constructivism
to the Brouwer fixed point theorem and other theorems regarding values of continuous functions on the reals. The fixed point theorem in turn implies
Constructive_set_theory
sequences, and structures. recursion theorem 1. Master theorem (analysis of algorithms) 2. Kleene's recursion theorem recursive definition A definition
Glossary_of_logic
Morse–Kelley set theory Kleene–Brouwer ordering The Kleene–Brouwer ordering is a total order on the finite sequences of ordinals Kleene hierarchy A classification
Glossary_of_set_theory
School of thought in philosophy of mathematics
numerals – each number has its predecessor as a subset. Kleene observes the following. (Kleene's assumptions (1) and (2) state that 0 has property P and
Logicism
Logic principle
for the (current) population of this village. Identity of indiscernibles Kleene equality Type theory Univalence axiom The Univalent Foundations Program
Extensionality
Particular class of sets which can be described entirely in terms of simpler sets
1 C K {\displaystyle \omega _{1}^{\mathrm {CK} }} stands for the Church–Kleene ordinal), and conversely any subset of ω {\displaystyle \omega } that belongs
Constructible_universe
Brouwer presents the Brouwer fixed-point theorem. 1912 – Josip Plemelj publishes simplified proof for the Fermat's Last Theorem for exponent n = 5. 1915 –
Timeline_of_mathematics
Transforming a function in such a way that it only takes a single argument
Kenneth (eds.). "Some Philosophical Aspects of Combinatory Logic". The Kleene Symposium: Proceedings of the Symposium Held June 18-24, 1978 at Madison
Currying
Informal set theories
and in a review by Laszlo Kalmar (Laszlo Kalmar (1946). "The Paradox of Kleene and Rosser". Journal of Symbolic Logic. 11 (4): 136.). The term was later
Naive_set_theory
Concept in logic
an abstract formal system. Revue philosophique de Louvain 50, 251–269. Kleene, S. C. (1967). Mathematical Logic. Reprinted 2002, Dover. ISBN 0-486-42533-9
Substitution_(logic)
Number used for counting
at either 0 or 1 and continue in their familiar fixed order – 1, 2, 3, and so on – with no end point. Each natural number labels a specific position in
Natural_number
Academic subfield of computer science
computation were Ramon Llull, Alonzo Church, Kurt Gödel, Alan Turing, Stephen Kleene, Rózsa Péter, John von Neumann and Claude Shannon. Automata theory is the
Theory_of_computation
Various systems of symbolic logic
provability), are Kurt Gödel’s dialectica interpretation, Stephen Cole Kleene’s realizability, Yurii Medvedev’s logic of finite problems, or Giorgi Japaridze’s
Intuitionistic_logic
Symbolic description of a mathematical object
expression, the lambda expression, was introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation. The lambda operators (lambda
Expression_(mathematics)
Mathematical function that can be computed by a program
term "computable", a distinction stemming from a 1934 discussion between Kleene and Gödel. For example, one can formalize computable functions as μ-recursive
Computable_function
Apparent contradiction in metamathematics
Curry's paradox List of self–referential paradoxes Kleene–Rosser paradox List of paradoxes Löb's theorem Ordinal definable set, a set-theoretic concept of
Richard's_paradox
Model of concurrent computation
generalization of the Church-Turing-Rosser-Kleene thesis [Kleene 1943]: A consequence of the above theorem is that a finite actor can nondeterministically
Actor_model
Logic with discrete truth values
Emil Leon Post introduced further truth degrees in 1921. Stephen Cole Kleene and Ulrich Blau expanded the three-valued logic system of Łukasiewicz, for
Finite-valued_logic
Many-valued logic in which truth values comprise a continuous range
arising in connection with the semantic paradoxes — by the schemes of Frege, Kleene, van Fraassen, or perhaps some other." Kripke, Saul (1975). "Outline of
Infinite-valued_logic
In logic, defining a new symbol
description Epsilon calculus Extension by new constant and function names S. C. Kleene (1952), Introduction to Metamathematics, D. Van Nostrand E. Mendelson (1997)
Extension_by_definition
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