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.

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

* Hilbert's program

In continuation of his " program " with which he challenged the mathematics community in 1900, at a 1928 international conference David Hilbert asked three questions, the third of which became known as " Hilbert's ".

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

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's incompleteness theorems and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

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.

* Richard Zach, 2006, " Hilbert's program then and now ", in Philosophy of Logic, Dale Jacquette ( ed.

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.

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.

( See Hilbert's program.

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.

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.

1931: Publication of Gödel's incompleteness theorems, showing that essential aspects of Hilbert's program could not be attained.

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.

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.

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.

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

David Hilbert was the first to invoke the term " metamathematics " with regularity ( see Hilbert's program ).

* 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.

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 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 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.

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.

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.

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 ).

At the start of the twentieth century mathematicians took up the axiomatic method, strongly influenced by David Hilbert's example.