The is related to Hilbert's tenth problem, which asks for an algorithm to decide whether Diophantine equations have a solution.

# REDIRECT Hilbert's tenth problem

Matiyasevich's completion of the MRDP theorem settled Hilbert's tenth problem.

Hilbert's tenth problem was to find a general algorithm which can decide whether a given Diophantine equation has a solution among the integers.

While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the ( philosophical ) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.

Hilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900.

The unsolvability of Hilbert's tenth problem is a consequence of the surprising fact that the

Because there exists a recursively enumerable set that is not computable, the unsolvability of Hilbert's tenth problem is an immediate consequence.

Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: " to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers.

In 1970, Yuri Matiyasevich proved ( using results of Julia Robinson ) Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution ; this problem asked whether there is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers.

1970: Hilbert's tenth problem is proven unsolvable: there is no recursive solution to decide whether a Diophantine equation ( multivariable polynomial equation ) has a solution in integers.

Hilbert's tenth problem asks for an algorithm to determine whether a Diophantine equation has any solutions in integers.

He is best known for his negative solution of Hilbert's tenth problem, presented in his doctoral thesis, at LOMI ( the Leningrad Department of the Steklov Institute of Mathematics ).

In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto, that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century.

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent.

* Hilbert's second problem

# REDIRECT Hilbert's fifth problem

Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples.

He also published papers on mathematical logic, and solved a special case of Hilbert's fifth problem.

( See Hilbert's second problem.

Work on Hilbert's 10th problem led in the late twentieth century to the construction of specific Diophantine equations for which it is undecidable whether they have a solution, or even if they do, whether they have a finite or infinite number of solutions.

More fundamentally, Hilbert's first problem was on the continuum hypothesis.

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number, that is not zero or one, and b is an irrational algebraic number, is a < sup > b </ sup > necessarily transcendental?

In Hilbert's sixth problem, he challenged researchers to find an axiomatic basis to all of physics.

Church and Turing independently demonstrated that Hilbert's Entscheidungsproblem ( decision problem ) was unsolvable, thus identifying the computational core of the incompleteness theorem.

While he is best known for the Kolmogorov – Arnold – Moser theorem regarding the stability of integrable Hamiltonian systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result — the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.

While a student of Andrey Kolmogorov at Moscow State University and still a teenager, Arnold showed in 1957 that any continuous function of several variables can be constructed with a finite number of two-variable functions, thereby partially solving Hilbert's thirteenth problem.

In fact, Smale's list contains some of the original Hilbert problems, including the Riemann hypothesis and the second half of Hilbert's sixteenth problem, both of which are still unsolved.

The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics.

Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided ( see " Modern viewpoints on the status of the problem ").

* Hilbert's second problem

On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

One of the important problems for logicians in the 1930s was David Hilbert's Entscheidungsproblem, which asked if there was a mechanical procedure for separating mathematical truths from mathematical falsehoods.

Hilbert, the first of two children of Otto and Maria Therese ( Erdtmann ) Hilbert, was born in the Province of Prussia-either in Königsberg ( according to Hilbert's own statement ) or in Wehlau ( known since 1946 as Znamensk ) near Königsberg where his father worked at the time of his birth.

Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and also a native of Königsberg.

Gordan, the house expert on the theory of invariants for the Mathematische Annalen, was not able to appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive.

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

In 1920 he proposed explicitly a research project ( in metamathematics, as it was then termed ) that became known as Hilbert's program.

In 1931 his incompleteness theorem showed that Hilbert's grand plan was impossible as stated.

In the early 20th century it was shaped by David Hilbert's program to prove the consistency of foundational theories.

It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier.

This list was compiled in the spirit of Hilbert's famous list of problems produced in 1900.

The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program.

Hilbert's program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency ( provided that they are in fact consistent ).

This relationship was discovered by David Hilbert who proved Nullstellensatz and several other important related theorems named after him ( like Hilbert's basis theorem ).

Hilbert's original question was more complicated: given any two tetrahedra T < sub > 1 </ sub > and T < sub > 2 </ sub > with equal base area and equal height ( and therefore equal volume ), is it always possible to find a finite number of tetrahedra, so that when these tetrahedra are glued in some way to T < sub > 1 </ sub > and also glued to T < sub > 2 </ sub >, the resulting polyhedra are scissors-congruent?

In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems.

One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.