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His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation, and his namesake Hilbert space plays an important part in quantum theory. In 1926, von Neumann showed that, if quantum states were understood as vectors in Hilbert ...
Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic. The value of Hilbert's Grundlagen was more methodological than substantive or pedagogical.
Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work. [d] [e] Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a ...
The Hilbert transform arose in Hilbert's 1905 work on a problem Riemann posed concerning analytic functions, [6] [7] which has come to be known as the Riemann–Hilbert problem. Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle. [8] [9] Some of his earlier work related to the Discrete Hilbert ...
[6] [7] Every Hilbert system is an axiomatic system, which is used by many authors as a sole less specific term to declare their Hilbert systems, [8] [9] [10] without mentioning any more specific terms. In this context, "Hilbert systems" are contrasted with natural deduction systems, [3] in which no axioms are used, only inference rules.
Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of ...
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous ) functions of two arguments .
This problem is more commonly called the Riemann–Hilbert problem.It led to several bijective correspondences known as 'Riemann–Hilbert correspondences', for flat algebraic connections with regular singularities and more generally regular holonomic D-modules or flat algebraic connections with regular singularities on principal G-bundles, in all dimensions.