Solved, Unsolved and Unsolvable: The Status of Hilbert’s 23 Problems in Mathematics

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By Evelyn Lamb<br>June 18, 2026

In the 126 years since David Hilbert presented his famous 23 problems, mathematicians have made tremendous advancements. Sean McCabe for Simons Foundation; Portrait of David Hilbert by L. Reidemeister/Archives of the Mathematisches Forschungsinstitut Oberwolfach

In 1900, prominent mathematician David Hilbert presented a list of 23 mathematical challenges for the next century to the International Congress of Mathematicians. These problems have become milestones in the field’s progress. Lectures at the 2026 ICM in Philadelphia will present exciting new research into two of these questions, the first and the sixth.

Hilbert’s problems aren’t so much finish lines at the end of a race as they are cracked doors inviting mathematicians to see what’s in the next room. For many of the problems, it is difficult to say definitively whether they are resolved or still open. Often, a question is resolved in one context but remains open in others. Sometimes a resolution opens the door to asking the same question in a different mathematical setting. This open-endedness is part of the problems’ ongoing appeal to mathematicians.

1. The continuum hypothesis. This is the question of whether there are sizes of infinity between the infinity of the counting (whole) numbers and the infinity of the real numbers. From the work of Kurt Gödel and Paul Cohen, mathematicians know that the continuum hypothesis is independent of the usual axioms for set theory, but different axiomatic viewpoints still leave room for further work. At the 2026 ICM, David Aspero and Ralf Schindler will present their recent work on the problem. Status: It’s complicated

2. The compatibility of the axioms of arithmetic. Hilbert wished to situate the rules of arithmetic in an airtight axiomatic system that does not permit contradictions. Again, Gödel burst his bubble. His incompleteness theorems proved that Peano arithmetic, a common axiomatic basis for arithmetic, cannot be proved consistent using only its own axioms. There is some debate about whether this is an adequate resolution of the problem. Status: It’s complicated

3. Equidecomposability. Given any two polyhedra of different volumes, can one be cut into a finite number of pieces and reassembled to be congruent to the other? This relationship is sometimes called scissors congruence. The analogous problem in two dimensions is possible, but it turns out not to be possible in three. This was the first Hilbert problem to fall. In 1900 (before the written version of Hilbert’s lecture had even been published), Hilbert’s student Max Dehn produced a counterexample. Status: Resolved

Proving that two polygons have the same area can be as easy as cutting them up and rearranging the pieces, as shown in this scissor congruent square and triangle. A counterexample proved that this equidecomposability is not necessarily true for polyhedra of equal volume in three dimensions. Lucy Reading-Ikkanda/Simons Foundation; Source: Wolfram Mathworld

4. The straight line as the shortest distance between two points. This question deals with exotic geometries in which the straight line is the shortest distance between two points but other properties of standard Euclidean geometry do not hold. This question is now considered too vague to have a clear resolution, but it has inspired investigation. Status: Too vague

5. Lie groups. These groups are algebraic objects that describe continuous transformations. In his original formulation, Norwegian mathematician Sophus Lie assumed that these transformations were also differentiable: that is, that the continuous transformations they described could be analyzed using the tools of calculus. Hilbert wondered whether differentiability was a necessary assumption or whether it could be proved from the other properties of Lie groups. In the early 1950s, Andrew Gleason, Deane Montgomery and Leo Zippin showed that the differentiability assumption is not necessary. (A different, more general interpretation of the problem is still unresolved.) Status: Resolved for some cases

6. The axiomatization of physics. Deriving solid mathematical foundations for physics will probably be a never-ending quest. Various fields have been axiomatized, from classical mechanics in 1903 to fluid dynamics in 2025 by Yu Deng, Zaher Hani and Xiao Ma. Status: Resolved for some cases

7. The transcendence of certain numbers. Most numbers are transcendental, meaning that they cannot be the solution to a polynomial equation, but proving any specific number to be transcendental is usually not an easy task. Hilbert wanted to show that ab must be transcendental when a is an algebraic number other than 0 and 1 and b is an irrational algebraic...