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.

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 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 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 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 tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

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.

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

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

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 ).

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

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.

: Foundations of Geometry ) published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms, substituting the traditional axioms of Euclid.

*' From Hilbert's Problems to the Future ', lecture by Professor Robin Wilson, Gresham College, 27 February 2008 ( available in text, audio and video formats ).

The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic.

Hilbert's paradox of the Grand Hotel is a mathematical veridical paradox ( a non-contradictory speculation that is strongly counter-intuitive ) about infinite sets presented by German mathematician David Hilbert ( 1862 – 1943 ).

The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of Transfinite Numbers.

* The novel White Light by mathematician / science fiction writer Rudy Rucker includes a hotel based on Hilbert's paradox, and where the protagonist of the story meets Georg Cantor.

Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency.

He started with the " betweenness " of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether ( or in what way ) these theorems answer Hilbert's second problem.

Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.

However, the question is still debated since in the literature there have been other such claims, largely based on different interpretations of Hilbert's statement of the problem given by various researchers.

Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

But doubtless the significance of Gödel's work to mathematics as a whole ( and not just to formal logic ) was amply and dramatically illustrated by its applicability to one of Hilbert's problems.

The " 24th problem " ( in proof theory, on a criterion for simplicity and general methods ) was rediscovered in Hilbert's original manuscript notes by German historian Rüdiger Thiele in 2000.

At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute.

He contributed to Hilbert's program in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic.

The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means.

However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is no smaller than total " number " of rooms.

The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics ( including its recognition as a discipline independent from mathematics ) seem to render both more remote and less important than in Hilbert's time.

This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means ( it was never made clear exactly what axioms were the " finitistic " ones, but whatever axiomatic system was being referred to, it was a ' weaker ' system than the system whose consistency it was supposed to prove ).

We may consider that Hilbert's program has been partially completed, so that the crisis is essentially resolved, satisfying ourselves with lower requirements than Hibert's original ambitions.

Today's student of geometry has axioms other than Euclid's available: see Hilbert's axioms and Tarski's axioms.

The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore fortuitous, rather than being an object of Hilbert's spectral theory.

It is considered by some to be a better formulation of Hilbert's fifth problem, than the characterisation in the category of topological groups of the Lie groups often cited as a solution.

The value of Hilbert's Grundlagen was more methodological than substantive or pedagogical.

This dealt a final blow to the heart of Hilbert's program, the hope that consistency could be established by finitistic means ( it was never made clear exactly what axioms were the " finitistic " ones, but whatever axiomatic system was being referred to, it was a weaker system than the system whose consistency it was supposed to prove ).

The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i. e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements.

Unlike some other modern axiomatizations, such as Birkhoff's and Hilbert's, Tarski's axiomatization has no primitive objects other than points, so a variable or constant cannot refer to a line or an angle.