Definition:Hilbert 23

Definition
The Hilbert 23 is a list of twenty-three problems in mathematics published by during 1900.

They are as follows:

2: Consistency of Axioms of Mathematics
Proof that the axioms of mathematics are consistent.

3: Finite Dissection of Polyhedra
Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second?

4: Construction of all Metrics where Lines are Geodesics
Construct all metrics where lines are geodesics.

5: Whether Continuous Groups are Differential Groups
Are continuous groups automatically differential groups?

6: Axiomatize all of Physics
Axiomatize all of physics.

9: General Reciprocity Theorem in Algebraic Number Field
Find the most general law of the Reciprocity Theorem in any algebraic number field.

10: Algorithm to determine whether Polynomial Diophantine Equation has Integer Solution
There is no algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.

11: Quadratic Forms with Algebraic Numerical Coefficients
Solving quadratic forms with algebraic numerical coefficients.

12: Extension of Kronecker-Weber Theorem to any base Number Field
Extend the Kronecker-Weber Theorem on abelian extensions of the rational numbers to any base number field.

13: Solution of 7th Degree Equations using Two Parameter Functions
Solve all 7th degree equations using functions of two parameters.

14: Proof of Finiteness of certain Complete Systems of Functions
Proof of the finiteness of certain complete systems of functions.

15: Rigorous foundation of Schubert's Enumerative Calculus
Rigorous foundation of Schubert's Enumerative Calculus.

16: Topology of Algebraic Curves and Surfaces
Topology of algebraic curves and surfaces.

17: Definite Rational Function as Quotient of Sums of Squares
Expression of definite rational function as quotient of sums of squares.

18a: Existence of Non-Regular Space-Filling Polyhedron
There exists a non-regular, space-filling polyhedron.

18b: Densest Sphere Packing
What is the densest Sphere Packing?

19: Solutions of Lagrangians are Analytic
Are the solutions of Lagrangians always analytic?

20: Existence of Solutions of Variational Problems with certain Boundary Conditions
Do all variational problems with certain boundary conditions have solutions?

21: Existence of Linear Differential Equation with prescribed Monodromic Group
Proof of the existence of linear differential equations having a prescribed monodromic group.

22: Uniformization of Analytic Relations by means of Automorphic Functions
Uniformization of analytic relations by means of automorphic functions.

23: Further Development of the Calculus of Variations
Further development of the calculus of variations.

There was originally going to be a 24th problem, on a criterion for simplicity and general methods in proof theory, but Hilbert decided not to include it, as it was (like numbers 4, 6, 16 and 23) too vague to ever be described as "solved".