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

Some of the axioms coincide, while some of the axioms in Moore's system are theorems in Hilbert's and vice-versa.

* Hilbert's axioms

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 axioms: Hilbert's axioms had the goal of identifying a simple and complete set of independent axioms from which the most important geometric theorems could be deduced.

* 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.

* Hilbert's axioms

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.

Hilbert's system consisting of 20 axioms < ref > a 21 < sup > st </ sup > axiom appeared in the French translation of Hilbert's Grundlagen der Geometrie according to most closely follows the approach of Euclid and provides the justification for all of Euclid's proofs.

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.

These do not provide a resolution to Hilbert's second question, however, because someone who doubts the consistency of Peano arithmetic is unlikely to accept the axioms of set theory ( which is much stronger ) to prove its consistency.

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.

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 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.

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

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.

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.

But these meta-mathematical proofs cannot be represented within the arithmetical calculus ; and, since they are not finitistic, they do not achieve the proclaimed objectives of Hilbert's original program.

This became known as the Kronecker Jugendtraum ; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field, via Shimura's reciprocity law.

Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.

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.

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.

Although not very successful in that respect, the lambda calculus found early successes in the area of computability theory, such as a negative answer to Hilbert's Entscheidungsproblem.

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.

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.

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

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.

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 " 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.

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics.

Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system ; then his incompleteness theorems showed that this is unattainable.

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.

** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem

There is some historical evidence that this fact influenced Hilbert's thinking about the prospects for proof theory.

* it motivated the currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis.

After Hilbert's initial formulation, the later development of abstract Hilbert space and the spectral theory of a single normal operator on it did very much go in parallel with the requirements of physics ; particularly at the hands of von Neumann.

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.

This refuted Hilbert's assumption that a finitistic system could be used to prove the consistency of a stronger theory.

This coincides with the ordinal successor operation for finite cardinals, but in the infinite case they diverge because every infinite ordinal and its successor have the same cardinality ( a bijection can be set up between the two by simply sending the last element of the successor to 0, 0 to 1, etc., and fixing ω and all the elements above ; in the style of Hilbert's Hotel Infinity ).