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

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.

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.

* 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.

* 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.

Hilbert's geometry is mathematical, because it talks about abstract points, but in Field's theory, these points are the concrete points of physical space, so no special mathematical objects at all are needed.

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.

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.

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.

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.

The analogous statement about polyhedra in three dimensions, known as Hilbert's third problem, is false, as proven by Max Dehn in 1900.

Hilbert's original statement of his 12th problem is rather misleading: he seems to imply that the abelian extensions of imaginary quadratic fields are generated by special values of elliptic modular functions, which is not correct.

Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem.

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 ").