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Statement in mathematical logic
In mathematical logic, the diagonal lemma (also known as diagonalization lemma, self-reference lemma or fixed point theorem) establishes the existence
Diagonal_lemma
Proof in set theory
over Cantor's theory Diagonal lemma the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's
Cantor's_diagonal_argument
Theorem that arithmetical truth cannot be defined in arithmetic
wholly elementary except for the diagonalization that the diagonal lemma requires. The proof of the diagonal lemma is likewise surprisingly simple; for
Tarski's undefinability theorem
Tarski's_undefinability_theorem
Topics referred to by the same term
techniques, including: Cantor's diagonal argument, used to prove that the set of real numbers is not countable Diagonal lemma, used to create self-referential
Diagonalization
Theorem in category theory
important corollaries of this are: Cantor's theorem Cantor's diagonal argument Diagonal lemma Russell's paradox Gödel's first incompleteness theorem Tarski's
Lawvere's_fixed-point_theorem
Topics referred to by the same term
argument (the earliest) Cantor's theorem Russell's paradox Curry's paradox Diagonal lemma Gödel's first incompleteness theorem Tarski's undefinability theorem
Diagonal_argument
Limitative results in mathematical logic
directly, the existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
ones) and the diagonal lemma hold for Peano arithmetics; once these are established the proof can be easily formalized. Using the diagonal lemma, there is
Hilbert–Bernays-Löb provability conditions
Hilbert–Bernays-Löb_provability_conditions
Embedding of categories into functor categories
The Yoneda lemma is a fundamental result in category theory, a branch of mathematics. It is an abstract result on functors of the type morphisms into
Yoneda_lemma
Mathematician and philosopher (1906–1978)
with ZFC Axiom of constructibility Compactness theorem Condensation lemma Diagonal lemma Dialectica interpretation Ordinal definable set Slingshot argument
Kurt_Gödel
Theorem for proving more complex theorems
local lemma Nakayama's lemma Noether normalization lemma Poincaré's lemma Riesz's lemma Schur's lemma Schwarz's lemma Sperner's lemma Urysohn's lemma Vitali
Lemma_(mathematics)
Method in mathematical logic
two numbers, as well as to include some first-order logic.) Using the diagonal lemma, let ρ {\displaystyle \rho } be a formula such that T {\displaystyle
Rosser's_trick
Self-replicating program
quine in the strict programming sense. Computer programming portal Diagonal lemma Droste effect Fixed point combinator Self-modifying code Self-interpreter
Quine_(computing)
Condition for a mathematical function to map some value to itself
fixed-point theorem Caristi fixed-point theorem Diagonal lemma, also known as the fixed-point lemma, for producing self-referential sentences of first-order
Fixed-point_theorem
Mathematical proposition equivalent to the axiom of choice
Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for
Zorn's_lemma
β function lemma makes use of the Chinese remainder theorem. Gödel numbering for sequences Gödel's incompleteness theorems Diagonal lemma H. E. Rose,
Gödel's_β_function
Sentence, idea or formula that refers to itself
reference – Series of references where the last object references the first Diagonal lemma – Statement in mathematical logic Droste effect – Recursive visual effect
Self-reference
Self-reference paradox
contradiction that (K) is both not known and known. Since, given the diagonal lemma, every sufficiently strong theory will have to accept something like
Knower_paradox
Distinction between using a word and mentioning it
this concept appears in Gödel's incompleteness theorem, where the diagonal lemma plays a crucial role. Stanisław Leśniewski extensively employed this
Use–mention_distinction
Abhyankar's lemma Fundamental lemma (Langlands program) Five lemma Horseshoe lemma Nine lemma Short five lemma Snake lemma Splitting lemma Yoneda lemma Matrix
List_of_lemmas
Provability logic
arithmetic, then the existence of modal fixed points follows from the diagonal lemma. In addition to the existence of modal fixed points, we assume the following
Löb's_theorem
choice Zorn's lemma Boolean algebra (structure) Boolean-valued model Burali-Forti paradox Cantor's back-and-forth method Cantor's diagonal argument Cantor's
List of mathematical logic topics
List_of_mathematical_logic_topics
American philosopher and logician (1940–2022)
informal self-referential meaning, and this idea – manifested by the diagonal lemma – is the basis for Tarski's theorem that truth cannot be consistently
Saul_Kripke
Theorem in computability theory
lambda calculus for the same purpose as the first recursion theorem. Diagonal lemma a closely related result in mathematical logic. Ershov, Yuri L. (1999)
Kleene's_recursion_theorem
Concept in mathematical set theory
mathematics, particularly in set theory, Fodor's lemma (or the pressing-down lemma) states: Fodor's lemma—If κ {\displaystyle \kappa } is a regular, uncountable
Fodor's_lemma
since no number is identical with its successor. Since, given the diagonal lemma, every sufficiently strong theory will have to accept something like
Hilbert–Bernays_paradox
In linear algebra
In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic
Matrix_determinant_lemma
Self-describing sentence
2's, 6 3's, 3 4's, 1 5, 2 6's, 1 7, 2 8's, and 1 9. Quine (computing) Diagonal lemma Sallows, L., In Quest of a Pangram, Abacus, Vol 2, No 3, Spring 1985
Autogram
X, G acts k-transitively on X n. Burnside's lemma can be proved using the action of the twofold diagonal subgroup. Diagonalizable group Sahai, Vivek;
Diagonal_subgroup
Maximal proper filter
of free ultrafilters on any infinite set is implied by the ultrafilter lemma, which can be proven in ZFC. On the other hand, there exist models of ZF
Ultrafilter_on_a_set
matrix whose diagonal block is 1 {\displaystyle 1} and i j {\displaystyle ij} -th entry is s {\displaystyle s} . The name "Whitehead's lemma" also refers
Whitehead's_lemma
Game theory
program game to be given access to their own source code. By the diagonalization lemma, one can use quining to enable programs to refer to their source
Program_equilibrium
Relationship between two functors abstracting many common constructions
notion of a limit. Any limit functor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question), and
Adjoint_functors
P(κ)/INS a κ+-complete Boolean algebra, when equipped with diagonal intersections. Club set Fodor's lemma Thomas Jech, Set Theory, The Third Millennium Edition
Diagonal_intersection
Collection of maps which give the same result
include those typically given for the five lemma, the snake lemma, the zig-zag lemma, and the nine lemma. In higher category theory, one considers not
Commutative_diagram
Result in mathematics and set theory
In mathematical logic, the Mostowski collapse lemma, also known as the Shepherdson–Mostowski collapse, is a theorem of set theory introduced by Andrzej
Mostowski_collapse_lemma
Generalization of algebraic spaces or schemes
groupoids "Lemma 92.10.11 (045G)—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-08-29. "Section 78.5 (046I): Bootstrapping the diagonal—The Stacks
Algebraic_stack
Mathematical concept in homotopy theory
In mathematics, specifically in homotopy theory, Ken Brown's lemma gives a sufficient condition for a functor on a category of fibrant objects to preserve
Ken_Brown's_lemma
Recursive integer sequence
{2n+1}{n}}\,,} which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurrence relations C 0 = 1
Catalan_number
Mathematical study of linear operators
operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward
Operator_theory
Central object of study in category theory
completely known and easy to describe; this is the content of the Yoneda lemma. Saunders Mac Lane, one of the founders of category theory, is said to have
Natural_transformation
Concepts from linear algebra
entries only along the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider
Eigenvalues_and_eigenvectors
Result in mathematical economics on existence of a non-negative equilibrium output vector
Kotelyanskiĭ, as it is referred to by Felix Gantmacher as Kotelyanskiĭ lemma. Diagonally dominant matrix Perron–Frobenius theorem Sylvester's criterion Hawkins
Hawkins–Simon_condition
alternative logical systems that deviate from classical logic. diagonalization lemma A lemma used in the proof of Gödel's incompleteness theorems, stating
Glossary_of_logic
In mathematics, invertible homomorphism
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Isomorphism
Category with direct sums and certain types of kernels and cokernels
stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions
Abelian_category
Branch of mathematical logic
weak weak Kőnig's lemma if and only if for every set X there is a set Y that is 1-random relative to X. DNR (short for "diagonally non-recursive") adds
Reverse_mathematics
Formula for the derivative of a matrix determinant
(A)\,dA)_{jj}=\operatorname {tr} (\operatorname {adj} (A)\,dA).\ \square } Lemma 1. det ′ ( I ) = t r {\displaystyle \det '(I)=\mathrm {tr} } , where det
Jacobi's_formula
Most general completion of a commutative square given two morphisms with same codomain
case E1 × E2 is a fiber bundle over B × B, and pulling back along the diagonal map B → B × B gives a space homeomorphic (diffeomorphic) to E1 ×B E2, which
Pullback_(category_theory)
Mapping between categories
morphism to itself. The identity functor is an endofunctor. Diagonal functor The diagonal functor is defined as the functor from D {\displaystyle D} to
Functor
Mathematical category
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Topos
Characterizes the diagonal of a Hermitian matrix with given eigenvalues
Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations
Schur–Horn_theorem
In category theory, a branch of mathematics, the diagonal functor C → C × C {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
Diagonal_functor
Impossibility result for ranked-choice voting systems
lemma statement, and let y=Best(Pz,Menu2(x)). Assume by contradiction that Px prefers z to y. By Lemma 1, f(Px,Pz)=Best(Pz, Menu2(x)) = y. By Lemma 3
Gibbard–Satterthwaite_theorem
Collection of objects and morphisms
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Category_(mathematics)
Map (arrow) between two objects of a category
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Morphism
Mathematical lemma
In axiomatic set theory, the Rasiowa–Sikorski lemma named after Helena Rasiowa and Roman Sikorski is one of the most fundamental facts used in the technique
Rasiowa–Sikorski_lemma
Lemma in constructibility theory
In set theory, a branch of mathematics, the condensation lemma is a result about sets in the constructible universe. It states that if X is a transitive
Condensation_lemma
Statement in mathematical combinatorics
either case the proof is complete. Lemma 1 implies that any R(r,s) is finite. The right hand side of the inequality in Lemma 2 expresses a Ramsey number for
Ramsey's_theorem
Construction in category theory
non-empty finite sets is non-empty. This is a generalization of Kőnig's lemma in graph theory and may be proved with Tychonoff's theorem, viewing the
Inverse_limit
Most general completion of a commutative square given two morphisms with same domain
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Pushout_(category_theory)
simple region Heine–Borel theorem Intermediate value theorem Itô's lemma Kőnig's lemma Kőnig's theorem (set theory) Kőnig's theorem (graph theory) Lagrange's
List_of_mathematical_proofs
forms Arithmetic geometry: Andrew Wiles (1995) modularity lifting Hensel's lemma Monad (functional programming) uses map functional to lift simple operators
Lift_(mathematics)
Type of matrix representation
The existence of a polar decomposition is a consequence of Douglas' lemma: Lemma—If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then
Polar_decomposition
General theory of mathematical structures
and as morphisms the natural transformations of such functors. The Yoneda lemma is one of the most famous basic results of category theory; it describes
Category_theory
Injective homomorphism
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Monomorphism
Special case of colimit in category theory
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Direct_limit
The Moschovakis coding lemma is a lemma from descriptive set theory involving sets of real numbers under the axiom of determinacy (the principle — incompatible
Moschovakis_coding_lemma
Theorem in linear algebra
B), then B = eiφ D AD−1 for some diagonal unitary matrix D (i.e. diagonal elements of D equals to eiΘl, non-diagonal are zero). If some power Aq is reducible
Perron–Frobenius_theorem
Mathematical concept
may be thought of as the category of all diagrams of shape J in C. The diagonal functor Δ : C → C J {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal
Limit_(category_theory)
Category admitting tensor products
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Monoidal_category
Collection of mathematical objects
Other equivalent forms are described in the following subsections. Zorn's lemma is an assertion that is equivalent to the axiom of choice under the other
Set_(mathematics)
Functors which are surjective and injective on hom-sets
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Full_and_faithful_functors
Type of category in category theory
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Cartesian_closed_category
Election result probability theorem
Bertrand's ballot theorem is related to the cycle lemma. They give similar formulas, but the cycle lemma considers circular shifts of a given ballot counting
Bertrand's_ballot_theorem
Applications of category theory
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Applied_category_theory
Result about when a matrix can be diagonalized
result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because
Spectral_theorem
Special objects used in (mathematical) category theory
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Initial_and_terminal_objects
Generalization of category theory
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Higher_category_theory
Overview of and topical guide to category theory
Abelian category Exact sequence Exact functor Snake lemma Nine lemma Five lemma Short five lemma Mitchell's embedding theorem Injective cogenerator Derived
Outline_of_category_theory
Theorem of matrix ranks
original matrix. Alternative names for this formula are the matrix inversion lemma, Sherman–Morrison–Woodbury formula or just Woodbury formula. However, the
Woodbury_matrix_identity
Type of category in category theory
projection morphisms, and ik will denote the injection morphisms. The diagonal morphism is the canonical morphism ∆: A → A ⊕ A, induced by the universal
Additive_category
f^{*}} and is denoted by f ∗ {\displaystyle f_{*}} . Ken Brown's lemma Ken Brown's lemma gives a sufficient condition for a functor to preserve weak equivalences
Glossary_of_category_theory
Category theory constructs
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Kan_extension
Proof by Alan Turing
requires the use of formal logic to prove a first lemma, followed by a brief word-proof of the second: Lemma 1: If S1 [symbol "0"] appears on the tape in some
Turing's_proof
Abstract mathematics relationship
Morphism Epi Mono Iso Zero Natural transformation Universal property Yoneda lemma Universal constructions Limits Terminal objects Products Equalizers Kernels
Equivalence_of_categories
Quadrilateral whose vertices lie on a circle
which E divides one diagonal equals that of the other diagonal. This is known as the intersecting chords theorem since the diagonals of the cyclic quadrilateral
Cyclic_quadrilateral
Surjective homomorphism
→ Y is not surjective, let y ∈ Y − fX. Since fX is closed, by Urysohn's Lemma there is a continuous function g1 : Y → [0, 1] such that g1 is 0 on fX and
Epimorphism
Mathematical category formed by reversing morphisms
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Opposite_category
Characterizing property of mathematical constructions
× C {\displaystyle {\mathcal {C}}\times {\mathcal {C}}} and define the diagonal functor Δ : C → C × C {\displaystyle \Delta :{\mathcal {C}}\to {\mathcal
Universal_property
Category whose objects and morphisms are inside a bigger category
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Subcategory
Sequence of homomorphisms such that each kernel equals the preceding image
snake lemma shows how a commutative diagram with two exact rows gives rise to a longer exact sequence. The nine lemma is a special case. The five lemma gives
Exact_sequence
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Elementary_topos
of Γ[a,b]. The generalisation of Morse's lemma to CAT(-1) spaces is often referred to as the Morse–Mostow lemma and can be proved by a straightforward generalisation
Busemann_function
Category theory
diagram Cone End Exponential Functor Adjoint functors Conservative Derived Diagonal Equivalence of categories Essentially surjective Exact Full and faithful
Kleisli_category
Matrix equal to its transpose
for every }}i,j,\quad a_{ji}=a_{ij}.} Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different
Symmetric_matrix
Generalization of category
n-category Doctrine (mathematics) Pseudofunctor String diagram 2-Yoneda lemma Pasting theorem Ehresmann 1965 Bénabou 1967 Kelly & Street 1974, § 1.2.
2-category
Generalization of a category
but space-valued (for example, for a correct formulation of the Yoneda lemma). The homotopy hypothesis says that one can take an ∞-groupoid, concretely
Quasi-category
Indexed collection of objects and morphisms in a category
each diagram to its colimit. The universal functor of a diagram is the diagonal functor; its right adjoint is the limit and its left adjoint is the colimit
Diagram_(category_theory)
Generalized object in category theory
category C × C . {\displaystyle \mathbf {C} \times \mathbf {C} .} The diagonal functor Δ : C → C × C {\displaystyle \Delta :\mathbf {C} \to \mathbf {C}
Product_(category_theory)
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