The List of Hilbert’s Twenty-Three Problems

David Hilbert was one of the most influential mathematicians of the 19th and early 20th centuries.

On August 8, 1900, Hilbert attended a conference at the Sorbonne, Paris, and presented a list of unsolved mathematics problems at the time. Hilbert presented ten of the problems on that day. To show his support to Cantor, Hilbert chose a question about Cantor’s continuity hypothesis to be his first. Then, he would add thirteen more questions.

The complete list of 23 problems was published later in the American Mathematical Society Bulletin by Mary Frances Winston Newson.

Hilbert’s First Problem

Continuum Hypothesis: The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers) To mathematicians, all infinities are not the same. The infinity of the counting numbers — 1, 2, 3, … — is smaller than…

Hilbert’s Second Problem

The Compatibility of the Standard Axioms of Arithmetic: Prove that the axioms of arithmetic are consistent. Hilbert’s second problem was to prove that arithmetic is consistent, that is, that no contradictions arise from the basic assumptions he had put forth…

Hilbert’s Third Problem

Hilbert’s third problem, the problem of defining volume for polyhedra, is a story of both threes and infinities.

Hilbert’s Fourth Problem

In mathematics, Hilbert’s fourth problem in the 1900 list of Hilbert’s problems is a foundational question in geometry.

Hilbert’s Fifth Problem

Understanding Lie Groups: Are continuous groups automatically differential groups Hilbert’s fifth problem concerns Lie groups, which are algebraic objects that describe continuous transformations. Hilbert’s question is whether Lie’s original framework, which assumes that certain functions are differentiable, works without the assumption of differentiability….

Hilbert’s Sixth Problem

The Axiomatization of Physics: Mathematical treatment of the axioms of physics (a) axiomatic treatment of probability with limit theorems for foundation of statistical physics (b) the rigorous theory of limiting processes “which lead from the atomistic view to the laws of motion of continua”…

Hilbert’s Seventh Problem

Irrationality and Transcendence of Certain Numbers: Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b  A number is called algebraic if it can be the zero of a polynomial with rational coefficients. For example, 2 is a zero of the polynomial x − 2, and √2 is a…

Hilbert’s Ninth Problem

Hilbert’s ninth problem is on algebraic number fields, extensions of the rational numbers to include, say, √2 or certain complex numbers.

Hilbert’s Tenth Problem

Hilbert’s tenth problem concerns finding an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Polynomial equations in a finite number of variables with integer coefficients are known as Diophantine equations. Equations like x2 − y3 = 7 and x2 +…

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