# Definition:Hilbert 23

## Definition

The **Hilbert 23** is a list of $23$ at-the-time unsolved problems in mathematics published by David Hilbert during $1900$.

They are as follows:

### 1: The Continuum Hypothesis

There is no set whose cardinality is strictly between that of the integers and the real numbers.

Symbolically, the **continuum hypothesis** asserts:

- $\aleph_1 = \mathfrak c$

where:

- $\mathfrak c$ denotes the cardinality of the continuum
- $\aleph_1$ denotes Aleph One.

### 2: Consistency of Axioms of Mathematics

Proof that the axioms of mathematics are consistent.

Consistency of Axioms of Mathematics

### 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?

Let $A$ and $B$ be polyhedra with the same volume.

Then it is not necessarily the case that there exists a dissection of $A$ into finitely many components that may be reassembled to form $B$.

### 4: Construction of all Metrics where Lines are Geodesics

Construct all metrics where lines are geodesics.

Construction of all Metrics where Lines are Geodesics

### 5: Whether Continuous Groups are Differential Groups

Are continuous groups automatically differential groups?

Whether Continuous Groups are Differential Groups

### 6: Axiomatize all of Physics

Axiomatize all of physics.

Mathematical treatment of the axioms of physics:

- $(1): \quad$ An axiomatic treatment of probability with limit theorems for foundation of statistical physics

- $(2): \quad$ The rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua".

### 7: The Gelfond-Schneider Theorem

Let $\alpha$ and $\beta$ be algebraic numbers (possibly complex) such that $\alpha \notin \set {0, 1}$.

Let $\beta$ be irrational.

Then any value of $\alpha^\beta$ is transcendental.

### 8a: The Riemann Hypothesis

All the nontrivial zeroes of the analytic continuation of the Riemann zeta function $\zeta$ have a real part equal to $\dfrac 1 2$.

### 8b: The Goldbach Conjecture

Every even integer greater than $2$ is the sum of two primes.

### 8c: The Twin Prime Conjecture

It is conjectured that there exist infinitely many pairs of twin primes: that is, primes which differ by $2$.

### 9: General Reciprocity Theorem in Algebraic Number Field

Find the most general law of the Reciprocity Theorem in any algebraic number field.

General Reciprocity Theorem in 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.

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.

Extension of Kronecker-Weber Theorem to any base Number Field

### 13: Solution of 7th Degree Equations using Two Parameter Functions

Solve all $7$th degree equations using functions of two parameters.

Solution of 7th Degree Equations using Two Parameter Functions

### 14: Proof of Finiteness of certain Complete Systems of Functions

Proof of the finiteness of certain complete systems of functions.

Proof of Finiteness of certain Complete Systems of Functions

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

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.

Topology of Algebraic Curves and Surfaces

### 17: Definite Rational Function as Quotient of Sums of Squares

Expression of a definite rational function as quotient of sums of squares.

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.

Existence of Non-Regular Space-Filling Polyhedron

### 18b: Kepler's Conjecture (Densest Sphere Packing)

The densest packing of identical spheres in space is obtained when the spheres are arranged with their centers at the points of a face-centered cubic lattice.

This obtains a density of $\dfrac \pi {3 \sqrt 2} = \dfrac \pi {\sqrt {18} }$:

- $\dfrac \pi {\sqrt {18} } = 0 \cdotp 74048 \ldots$

### 19: Solutions of Lagrangian are Analytic

Are the solutions of a Lagrangian always analytic?

Solutions of Lagrangian are Analytic

### 20: Existence of Solutions of Variational Problems with certain Boundary Conditions

Do all variational problems with certain boundary conditions have solutions?

Existence of Solutions of Variational Problems with certain Boundary Conditions

### 21: Existence of Linear Differential Equation with prescribed Monodromic Group

Proof of the existence of linear differential equations having a prescribed monodromic group.

Existence of Linear Differential Equation with prescribed Monodromic Group

### 22: Uniformization of Analytic Relations by means of Automorphic Functions

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.

Further Development of the Calculus of Variations

## Historical Note

The **Hilbert 23** were delivered by David Hilbert in a famous address at Paris in $1900$.

He considered them to be the oustanding challenges to mathematicians in the future.

There was originally going to be a $24$th 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".

## Sources

- 1902: David Hilbert:
*Mathematical Problems*(*Bull. Amer. Math. Soc.***Vol. 8**,*no. 10*: pp. 437 – 479)

- (translated by Mary Winston Newson from "Mathematische Probleme")

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**Hilbert's problems**