# Hilbert's 20 axioms of the Euclidean geometry

• MathematicalPhysicist
In summary, Hilbert divides the axioms into five groups: seven axioms of connection, five axioms of order, one axiom of parallels, six axioms of congruence, and the axiom of continuity. These axioms define the properties of points, lines, and angles in a geometric system. They also introduce the concept of congruence and the division of a plane into two regions.

#### MathematicalPhysicist

Gold Member
what are they?

Originally posted by loop quantum gravity
what are they?

In his book The Foundations of Geometry (Open Court reprint 1965), Hilbert divides the axioms into five groups.

I Seven Axioms of Connection
I.1 Two distinct pioints always determine a straight line.
I.2 Any two points of a line completely determine that line.
I.3 Three points not situated in the same straight line always completely determine a plane.
I.4 Any three points of a plane, which do not lie in the same straight line, completely determine that plane.
I.5 If two points of a straight line lie in a plane a then all the points of that line lie in a.
I.6 If two planes have a point in common, then they have at least a second point in common.
I.7 Upon every line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.

II. Five Axioms of Order
II.1 If A, B, and C are points of a straight line and B lies between A and C, then B lies also between C and A.
II.2 If A and C are two points on a straight line thenthese exists at least one point B lying between A and C and at least one point D so situated that C lies between A and D.
II.3 Of three points situated on a straight line, there is always one and only one which lies between the other two.
II.4 For any four points of a straight line, the names A, b, c, and D can always be assigned to them in such a way that B shall lie between A and C and also between A and D, and furthermore that C shall lie between A and D and also between B and D.

Definition. The system of two points A and B on a straight line will be called a segment, denoted by AB or BA. The points between A and B are called the points of the segment AB. All other points of the line are referred to as points lying outside the segment AB.

II.5. Let A, B, C be three points not lying in the same straight line and let a be a straight line in the same plane as A, B, C not passing through any of the points A, B, or C. Then if the straight line a passes through a point of the segment AB, it will also pass through either if the segment BC or a point of the segment AC.

III One Axiom of Parallels
In a plane there can be drawn through any point A, lying outside a straight line a, one and only one straight line which does not intersect the line a. This straight line is called the parallel to a through the given point A.

IV Six Axioms of Congruence
IV.1 If A, B are two points on a straight line a, and A' is a point on the same or another straight line a', then upon a given side of A' on the stright line a' we can always find one and only one point B' so that the segment AB (or BA) is congruent to the segment A'B'. Every segment is congruent to itself.
IV.2 If a segment AB is congruent to the segment A'B' and also to the segment A"B" then the segment A'B' is congruent to the segment A"B".
IV.3. Let AB and BC be two segments of a straight line a which have no points in common except for the point B and furthermore let A'B' and B'C' be two segments of the same or another line a' having likewise no points in common except for the point B'. Then if AB is congruent to A'B' and BC is congruent to B'C' we have AC congruent to A'C'.

Definitions
Let a be any arbitrary plane and h,k, any two distinct half rays lying in a and eminationg from the point O so as to form a part of two different straight lines. The system formed by h,k is termed the angle (h,k). From the previous axioms we can prove that the angle divides the plane a into two regions and that one of these, termen the interior of the angle, has the property that any to points in it form a segment which lies entirely within the region.

IV.4 Let an angle (h,k) be given in the plane a and let a straight line a' be given in the plane a'. Suppose also that in the plane a' a defineite side of the straight line a' is assigned. Denote by k' a half ray of this line eminating from a point O' on this line. Then in the plane a' there is one and only one half ray k' such that the angle (h,k) is congruent to the angle (h',k') and at the same time all the interior points lie in the assiged side if the straight line a'. This defines the congruence of angled. Every angle is congruent to itself.
IV.5 If angle (h,k) is congruent to angle (h'k') and to the angle (h",k") then angel (h',k') is congruent to angle (h",k").
IV.6 If in the two triangles ABC and A'B'C' the congruences AB congruent to A'B' AC congruent to A'C' angle BAC congruent to angle B'A'C' hold, then
angle ABC will be congruent to angle A'B'C' and angle ACB will be congruent to angle A'C'B'.

V. Axiom of Continuity
Let A1 be any point on a straight line between two arbitrarily chosen points A and B. Take the points A2, A3, A4, ... so that A1 lies between A and A2, A2 between A1 and A3, A3 between A2 and A4. etc. Moreover let the segments AA1, A1A2, A2A3, etc. be all equal to one another.
Then among this series of points there always exists a certain point An such that B lies between A and An.

That's the lot. I eliminated a few definitions like ray and triangle.

Hilbert's 20 axioms of Euclidean geometry are a set of fundamental principles that serve as the basis for the study and understanding of geometry. These axioms were developed by German mathematician David Hilbert in the late 19th and early 20th century and are still widely used today.

The 20 axioms can be divided into three categories: incidence axioms, order axioms, and congruence axioms. The incidence axioms describe the basic relationships between points, lines, and planes. The order axioms establish the concept of betweenness and the relationship of order between points on a line. The congruence axioms define the concept of congruence and the properties of congruent figures.

Some of the key axioms from each category include:

1. Incidence Axiom 1: Two distinct points determine a unique line.

2. Incidence Axiom 2: Any two distinct lines intersect in at most one point.

3. Order Axiom 1: If A, B, and C are three points on a line and B is between A and C, then AB + BC = AC.

4. Congruence Axiom 1: If two triangles have three sides of one equal respectively to three sides of the other, then the two triangles are congruent.

5. Congruence Axiom 2: If two triangles have two sides and the included angle of one equal respectively to two sides and the included angle of the other, then the two triangles are congruent.

These are just a few examples of the 20 axioms that form the foundation of Euclidean geometry. They may seem simple, but they are powerful tools that allow mathematicians to prove and discover new theorems and properties in geometry. Without these axioms, the study of Euclidean geometry would not be possible.

## 1. What are Hilbert's 20 axioms of Euclidean geometry?

Hilbert's 20 axioms are a set of statements that form the foundation of Euclidean geometry. These axioms describe the properties of points, lines, and planes, and how they relate to each other.

## 2. When were Hilbert's 20 axioms first proposed?

Hilbert first published his axioms in his book "Grundlagen der Geometrie" in 1899. However, he continued to refine and add to them throughout his career.

## 3. How do Hilbert's axioms differ from Euclid's axioms?

Euclid's axioms were more geometric in nature, focusing on constructions and shapes. Hilbert's axioms, on the other hand, were more abstract and algebraic, dealing with the relationships between geometric objects.

## 4. Are Hilbert's axioms still relevant in modern mathematics?

Yes, Hilbert's axioms are still considered an important foundation in modern mathematics. They are used in the study of geometry and in the development of other branches of mathematics, such as topology and algebraic geometry.

## 5. Are there any criticisms of Hilbert's axioms?

Yes, some mathematicians have criticized Hilbert's axioms for being too complex and difficult to understand. Others have argued that they are not sufficient to describe all aspects of geometry, and have proposed alternative sets of axioms.