* Hilbert's theorem ( differential geometry )

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

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

* Euclidean geometry, under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness and incidence.

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.

In Hilbert's axiomatization of geometry this statement is given as a theorem, but only after much groundwork.

Hilbert's Nullstellensatz ( German for " theorem of zeros ," or more literally, " zero-locus-theorem " – see Satz ) is a theorem which establishes a fundamental relationship between geometry and algebra.

We have already noted the example of analytic geometry, and more generally the field of algebraic geometry thoroughly develops the connections between geometric objects ( algebraic varieties, or more generally schemes ) and algebraic ones ( ideals ); the touchstone result here is Hilbert's Nullstellensatz which roughly speaking shows that there is a natural one-to-one correspondence between the two types of objects.

Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's ' new ' use of axiomatic method as a research tool.

For example, Hilbert's Nullstellensatz is a theorem which is fundamental for algebraic geometry, and is stated and proved in term of commutative algebra.

The notion arose from the theory of magnitudes of Ancient Greece ; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.

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

Therefore, this case is what usually is meant when talking about Hilbert's sixteenth problem in real algebraic geometry.

Taxicab geometry satisfies all of Hilbert's axioms ( a formalization of Euclidean geometry ) except for the side-angle-side axiom, as one can generate two triangles each with two sides and the angle between them the same, and have them not be congruent.

Other often-used axiomizations of plane geometry are Hilbert's axioms and Tarski's axioms.

An assignment of Halsted's led Moore to prove that one of Hilbert's axioms for geometry was redundant.

In mathematics, Hilbert's fourth problem in the 1900 Hilbert problems was a foundational question in geometry.

Hilbert's axioms for plane geometry number 16, and include Transitivity of Congruence and a variant of the Axiom of Pasch.

Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900.

There is, however, room to doubt whether Hilbert's own views were simplistically formalist in this sense.

* Hilbert's paradox of the Grand Hotel, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numbers.

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

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

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.

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?

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 ).

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

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.

This fact follows from the famous Hilbert's basis theorem named after mathematician David Hilbert ; the theorem asserts that if R is any Noetherian ring ( such as, for instance, ), R is also a Noetherian ring.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

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.

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 ").

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.

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's work had started logic on this course of clarification ; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s.

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics.

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

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.

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.

The third on Hilbert's list of mathematical problems, presented in 1900, is the easiest one.

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.

Since 1900, other mathematicians and mathematical organizations have announced problem lists, but, with few exceptions, these collections have not had nearly as much influence nor generated as much work as Hilbert's problems.

It thus became clear that the notion of mathematical truth can not be completely determined and reduced to a purely formal system as envisaged in Hilbert's program.

As previously mentioned, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program.

There were 23 problems on David Hilbert's famous list of unsolved mathematical problems, presented to the International Congress of Mathematicians in Paris in 1900.

David Hilbert's work on the question of the finite generation of the algebra of invariants ( 1890 ) resulted in the creation of a new mathematical discipline, abstract algebra.

Hilbert's seventh problem is one of David Hilbert's list of open mathematical problems posed in 1900.

Hilbert's eighth problem is one of David Hilbert's list of open mathematical problems posed in 1900.

Many current lines of research in mathematical logic, proof theory and reverse mathematics can be viewed as natural continuations of Hilbert's original program.