In mathematical logic, the diagonal lemma or fixed point theorem establishes the existence of self-referential sentences in certain formal theories of the natural numbers -- specifically those theories that are strong enough to represent all computable functions.

The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's indefinability theorem.

The diagonal lemma applies to theories capable of representing all primitive recursive functions.

The diagonal lemma states that there is a sentence φ such that φ ↔ ψ (< u >#( φ )</ u >) is provable in T.

The diagonal lemma is closely related to Kleene's recursion theorem in computability theory, and their respective proofs are similar.

The lemma is called " diagonal " because it bears some resemblance to Cantor's diagonal argument.

The terms " diagonal lemma " or " fixed point " do not appear in Kurt Gödel's epochal 1931 article, or in Tarski ( 1936 ).

Mendelson ( 1997, p. 204 ) believes that Carnap was the first to state that something like the diagonal lemma was implicit in Gödel's reasoning.

The resulting theorem applies to any formal language with negation, and with sufficient capability for self-reference that the diagonal lemma holds.

In particular, if A is a sentence of arithmetic then True ( g ( A )) holds in N if and only if A is true in N. Hence for all A, the Tarski T-sentence True ( g ( A )) ↔ A is true in N. But the diagonal lemma yields a counterexample to this equivalence, by giving a " Liar " sentence S such that S ↔ ¬ True ( g ( S )) holds.

The formal machinery of this proof is wholly elementary except for the diagonalization that the diagonal lemma requires.

The proof of the diagonal lemma is likewise surprisingly simple ; for example, it does not invoke recursive functions in any way.

Such languages are necessarily capable of enough self-reference for the diagonal lemma to apply to them.

He also emphasizes line within the work, by displaying the upward growth of the forest with the vertical and diagonal lines of the trunks.

A scattering of diagonal streets, many of them originally Indian trails, also cross the city.

For any rectangular section on a round tube, the diagonal measurement is also the diameter of the tube.

Sometimes a long diagonal slat or two are also implemented to prevent the door from skewing.

" It also contains the general statement of the Pythagorean theorem ( for the sides of a rectangle ): " The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together.

Gauss – Jordan elimination, an extension of this algorithm, reduces the matrix further to diagonal form, which is also known as reduced row echelon form.

* Vector symmetries ( also called diagonal symmetries ) mean the same transformation is applied on the two chiralities.

The foot pushes backward and also down, creating a diagonal force vector, which, in an efficient running style, is aimed squarely at the runner's centre of mass.

But Cantor's diagonal argument proves that the real numbers ( and therefore also the complex numbers ) are uncountable ; so the set of all transcendental numbers must also be uncountable.

He also showed how to solve systems of linear equations be reducing the matrix of their coefficents to diagonal form.

As a consequence of its diagonal movement, each bishop always remains on either the white or black squares, and so it is also common to refer to them as light-squared or dark-squared bishops.

Haidinger's brush may also be seen by looking at a white area on many LCD flat panel computer screens ( due to the polarization effect of the display ), in which case it is often diagonal.

Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers.

However, it demonstrates a powerful and general technique that has since been used in a wide range of proofs, also known as diagonal arguments by analogy with the argument used in this proof.

The great king Yu () tried to channel the water out to sea where then emerged from the water a turtle with a curious figure / pattern on its shell ; circular dots of numbers which were arranged in a three by three grid pattern such that the sum of the numbers in each row, column and diagonal was the same: 15, which is also the number of days in each of the 24 cycles of the Chinese solar year.

* inv is a two-sided inverse for m, i. e. if d: G → G × G is the diagonal map, and e < sub > G </ sub >: G → G is the composition of the unique morphism G → 1 ( also called the counit ) with e, then m ( id < sub > G </ sub > × inv ) d =

The third term on the right hand side of the expression for the matrix element of T < sub > n </ sub > ( the Born – Oppenheimer diagonal correction ) can approximately be written as the matrix of squared and, accordingly, is then negligible also.

In the special case that M is a normal matrix, which by definition must be square, the spectral theorem says that it can be unitarily diagonalized using a basis of eigenvectors, so that it can be written for a unitary matrix U and a diagonal matrix D. When M is also positive semi-definite, the decomposition is also a singular value decomposition.

The members were also strengthened by the addition of steelwork, and 18 new diagonal cable stays were fitted.

There is also much wasted space where the match data is inherently duplicated across the diagonal and most of the actual area of the plot is taken up by either empty space or noise, and, finally, dot-plots are limited to two sequences.

In domestic politics, Bethmann Hollweg's record was also mixed, and his policy of the " diagonal ", which endeavoured to maneuver between the Socialists and Liberals of the left and the right-wing nationalists of the right, only succeeded in alienating most of the German political establishment.

Thus, a solution requires that no two queens share the same row, column, or diagonal.

Our operation requires us to move the right diagonal to the bottom row, but that would lead to two rows of 3 elements, forbidden since we're counting partitions into distinct parts.

The first requires the player to eat falling beans by shooting Pyoro's tongue in an upward diagonal direction.

* Picture Tic-Tac-Toe which requires the reader to determine a common theme for each row, column, and diagonal of a 3x3 matrix of pictures.

To each paired vertex and diagonal point there corresponds a unique forward corner point, i.e., the corner on C reached first by proceeding along C from the vertex in the direction of increasing T.

We must now show that on some component of the graph there exist two points for which the corresponding diagonal points in the C-plane are on opposite sides of C.

Using the diagonal argument, it is possible to define a real number x, which is not equal to G ( n ) for any n. This means that there is a language L ' that defines x, which is undefinable in L.

Dark must place a piece with the dark side up on the board, in such a position that there exists at least one straight ( horizontal, vertical, or diagonal ) occupied line between the new piece and another dark piece, with one or more contiguous light pieces between them.

A diagonal relationship exists between the behavior of magnesium and scandium, just as there is between beryllium and aluminium.

These cases demonstrate a paradox not in the sense that they demonstrate a logical contradiction, but in the sense that they demonstrate a counter-intuitive result that is provably true: the situations " there is a guest to every room " and " no more guests can be accommodated " are not equivalent when there are infinitely many rooms ( an analogous situation is presented in Cantor's diagonal proof ).

The matrix M is positive definite if and only if there exists an upper triangular matrix, with strictly positive diagonal elements, such that.

In the context of classical mathematics, this is impossible, and the diagonal argument establishes that, although both sets are infinite, there are actually more infinite sequences of ones and zeros than there are natural numbers.

However, contrary to the diffusionist theory, it appears that there was simultaneously a structural evolution towards the pointed arch, for the purpose of vaulting spaces of irregular plan, or to bring transverse vaults to the same height as diagonal vaults.

Again, there is a kick as part of the ollie but unlike the kickflip it is directed forward and outwards away from the rider's toe side ( diagonal ), so that the last part of the foot to leave the board is the heel, hence the name.

Tests so far confirm that there are diagonal lines even when many numbers are plotted.

However, not all fixed-wing aircraft have tailplanes, such as those configured with canards, flying-wing aircraft, where there is no tail, and v-tail aircraft where the fin / rudder and tail-plane are combined to form two diagonal surfaces in a V layout.

Mclean was distinctive in that there were no streets between the diagonal line of lots along the tracks. Perhaps because of this, much of the business district developed along Morgan Street, which ran east-west just north of the park, or along Hamilton Street, which ran north-south, just west of the park.

The Burlington and Missouri built a line from Lincoln to Columbus, but stopped there ; for their diagonal route across Nebraska, they chose one that crossed the Union Pacific at Grand Island rather than Columbus.

More explicitly: For every symmetric real matrix A there exists a real orthogonal matrix Q such that D = Q < sup > T </ sup > AQ is a diagonal matrix.

The Cholesky decomposition is unique: given a Hermitian, positive-definite matrix A, there is only one lower triangular matrix L with strictly positive diagonal entries such that A

In fact, a given n-by-n matrix A is similar to a diagonal matrix ( meaning that there is a matrix X such that X < sup >- 1 </ sup > AX is diagonal ) if and only if it has n linearly independent eigenvectors.

The spectral theorem says that every normal matrix is unitarily similar to a diagonal matrix ( if AA < sup >*</ sup > = A < sup >*</ sup > A then there exists a unitary matrix U such that UAU < sup >*</ sup > is diagonal ).