Hilbert's problems

Hilbert's problems are 23 mathematical challenges presented by David Hilbert at the 1900 International Congress of Mathematicians. They shaped research across several fields, and some remain unresolved.

Continuum hypothesis: The claim that no set has cardinality strictly between the integers and the real numbers. Hilbert's first problem asks whether this hypothesis follows from accepted set-theoretic axioms.
David Hilbert: A German mathematician whose work shaped geometry, algebra, logic, and mathematical physics. He selected and presented the 23 challenges that define this list.
Hilbert's address on mathematical problems: David Hilbert's 1900 lecture outlining significant mathematical questions and research directions. The problems were introduced within an argument about how mathematical research advances.
Gödel's incompleteness theorems: Two theorems showing limits on what sufficiently expressive, consistent formal systems can prove about arithmetic. They transformed expectations surrounding foundational questions associated with Hilbert's program.
Hilbert's tenth problem: The problem of finding an algorithm to decide whether any polynomial equation with integer coefficients has an integer solution. Its negative solution turned one of Hilbert's challenges into a boundary of computation.
International Congress of Mathematicians: A major international gathering of mathematicians held periodically since 1897. Hilbert announced the problems at its 1900 meeting in Paris.
Axiomatic method: A method of developing a theory from explicit axioms through deductive reasoning. Several problems concern the foundations and consistency of mathematical axiom systems.
Hilbert's program: A project to secure mathematics by formalizing it and proving the consistency of its formal systems. The problems' foundational ambitions grew alongside this broader program.
Hilbert's seventh problem: The problem of proving that certain powers of algebraic numbers are transcendental. Gelfond and Schneider independently solved it, establishing the transcendence of numbers such as 2^√2.
Paris 1900 International Congress of Mathematicians: The second International Congress of Mathematicians, held in Paris in 1900. Hilbert delivered his address on mathematical problems at this congress.

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