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 Problem #1

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…

Hilbert’s Problem #2

The Compatibility of the Standard Axioms of Arithmetic: Prove that the axioms of arithmetic are consistent. Hilbert’s second problem was…

Hilbert’s Problem #3

Equidecomposability: Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that…

Hilbert’s Problem #4

The Straight Line as The Shortest Distance Between Points: Construct all metrics where lines are geodesics. Hilbert’s fourth problem is about what happens…

Hilbert’s Problem #5

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 Problem #6

The Axiomatization of Physics: Mathematical treatment of the axioms of physics (a) axiomatic treatment of probability with limit theorems for foundation of statistical physics…

Hilbert’s Problem #7

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…

Hilbert’s Problem #8

Problems of Prime Numbers: The Riemann hypothesis(“the real part of any non-trivial zero of the Riemann zeta function is ½”)and other prime number problems, among…

Hilbert’s Problem #9

Reciprocity Laws and Algebraic Number Fields: Find the most general law of the reciprocity theorem in any algebraic number field. Hilbert’s ninth problem is…

Hilbert’s Problem #10

Solvability of a Diophantine Equation: Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer…

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