# Hilbert’s First Problem

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 infinities are not the same. The infinity of the counting numbers — 1, 2, 3, … — is smaller than the infinity of all the real numbers. And there are towers of still greater infinities beyond the reals. Hilbert’s first problem, also known as the continuum hypothesis, is the statement that there is no infinity in between the infinity of the counting numbers and the infinity of the real numbers. In 1940, Kurt Gödel showed that the continuum hypothesis cannot be proved using the standard axioms of mathematics. In 1963, Paul Cohen showed it cannot be disproved, making the continuum hypothesis independent of the axioms of mathematics.

## Hilbert’s Twenty-second Problem

Hilbert’s Twenty-second Problem: Uniformization of analytic relations by means of automorphic functions Hilbert’s 22nd problem asks whether every algebraic or analytic curve — solutions to polynomial equations — can be written…

## Hilbert’s Twenty-first Problem

Hilbert’s Twenty-first Problem: Proof of the existence of linear differential equations having a prescribed monodromic group Hilbert’s 21st problem is about the existence of certain systems of differential equations with given singular points…

## Hilbert’s Twentieth Problem

Hilbert’s Twentieth Problem: Do all variational problems with certain boundary conditions have solutions Do solutions in general exist? The calculus of variations is a field concerned with optimizing certain types of functions called functionals.…

## Hilbert’s Problem #19

Are The Solutions of Regular Problems in The Calculus of Variations Always Necessarily Analytic: Are the solutions of regular problems in the calculus of variations always necessarily analytic Do solutions in general exist?…

## Hilbert’s Eighteenth Problem

Hilbert’s Eighteenth Problem: (a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? (b) What is the densest sphere packing? Hilbert’s 18th problem is a collection of several questions…

## Hilbert’s Seventh Problem

EHilbert’s Seventh Problem: Express a nonnegative rational function as quotient of sums of squares Some polynomials with inputs in the real numbers always take non-negative values; an easy example is x2 + y2. Hilbert’s 17th problem asks…

## Hilbert’s Sixteenth Problem

ToHilbert’s Sixteenth Problem: Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. Hilbert’s 16th problem is an expansion of grade school graphing questions.…

## Hilbert’s Fifteenth Problem

Hilbert’s Fifteenth Problem is the igorous foundation of Schubert’s enumerative calculus. Hilbert’s 15th problem is another question of rigor. He called for mathematicians to put Schubert’s enumerative calculus, a branch of…