Axioms of David Hilbert

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Over the years, mathematicians had been bothered by the unstated assumptions of Euclidean geometry, which has been the foundation of geometry for about 2000 years. One of these unstated assumptions is the concept of 'betweenness'. In the early 20th century, David Hilbert (1862-1943) reworked the axioms of Euclid's work. He expanded Euclid's original five axioms to 21 axioms in five groups. This article is based largely on David Hilbert's Foundations of Geometry1, 2nd English edition.

David Hilbert identified three basic sets of objects for geometry: points, lines, and planes. The element of line geometry is the point. The elements of plane geometry are the point and line. The elements of space geometry are points, lines, and planes.

Points, lines and planes have specific relationships. These relationship are described by the axioms of geometry:

blank spaceI. Axioms of Incidence,
blank spaceII. Axioms of Order,
blank spaceIII. Axioms of Congruence,
blank spaceIV. Axiom of Parallels, and
blank spaceV. Axioms of Continuity.

I. Axioms of Incidence

The word 'incidence' means one geometric object may contain another (see incidence, Dictionary.com). For example, a line may contain a point and, in fact, must be made up of at least two points. The axioms of incidence are:

I,1.

For every two points A and B, there exists a line that contains each of these points.

Axiom I,1 means if one picks any two points, there must be a line that connects those two points. To see the logic of this, try to think of two points in space that can not be connected by a line. In figure 1, click on the yellow points and drag them. Where ever the points are placed (except on top of each other), there is a line that passes through them.

Click on the blue points and drag them to change the figure.

Try to imagine two points which can not be connected with a line. Does it seem possible?
Manipulative 1 - Hilbert Axiom i1 Created with GeoGebra.
I,2.

For every two points A and B, there exists no more than one line that contains each of these points.

Axiom I.2 states that there is exactly one line that passes through two given points. In figure 2, click on one of the yellow points and move it right on top of the other yellow point. Do you see one line or two?

Click on the blue and red points and drag them to change the figure.

Move one blue point on top of the other. Do you see one line or two?
Manipulative 2 - Hilbert Axiom i2 Created with GeoGebra.
I,3.There exists at least two points on a line. There exists at least three points that do not lie on a line.

Try to imagine a line that has only one point. It is not a line, but a point. So each line must have at least two points. In figure 3, click on one of the yellow points and drag it on top of the other. Does the line still exist?

The second statement implies the existence of a plane. If there are at least three points that do not lie on a line, there must exist some mathematical construct to contain these three points. This construct is the plane. This implies that lines exist within planes.

Click on the yellow and blue points and drag them to change the figure.

Axiom I3 states a line contains at least two points. Try moving one yellow point on top of the other. What happens to the line segment.
Manipulative 3 - Hilbert Axiom i3 Created with GeoGebra.
I,4.

For any three points A, B, and C, there exists a plane α that contains the points A, B, and C. For every plane there exists a point that lies in the plane.

I,5.

For any three points A, B, and C that do not lie on one and the same line, there exists no more than one plane that contains each of the three points.

I,6.

If two points A and B of line a lie in a plane α, then every point of a lies in α.

I,7.

If two planes, α and β have a point A in common, then the have at least one point B in common.

I,8.

There exist at least four points that do not lie in a plane.

II. Axioms of Order

II,1.If a point B lies between a point A and a point C then the points A, B, and C are three distinct points and B also lies between C and A.
Click on the blue points and drag them the change the figure.

Note that point B is between point A and C, and the B is between points C and A and B is distinct from A and C.
Manipulative 9 - Hilbert Axiom ii1 Created with GeoGebra.
II,2.For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B.
Click on the blue points and drag them to change the figure.

There always exists a point B on line AC such that C lies between A and B.
Manipulative 10 - Hilbert Axiom ii2 Created with GeoGebra.
II,3.Of any three points on a line there exists no more than one that lies between the other two.
Click on the blue points and drag them to change the figure.

Can you arrange the points on the line so that two of them are between other points?
Manipulative 11 - Hilbert Axiom ii3 Created with GeoGebra.
II,4.Let A, B, and C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, or C. If the line a passes through a point of the segment AB, it also passes through a point of segment AC or through a point of the segment BC.
Click on the blue and red points and drag them to change the figure.

Is it possible for the red line to cross one of the sides of the triangle without crossing either of the other sides?
Manipulative 12 - Hilbert Axiom ii5 Created with GeoGebra.

III. Axioms of Congruence

III,1.If A and B are two points on line a, and A' is a point on the same or on another line a', then it is always possible to find another point B' on a given side of the line a' through A' such that segment AB is congruent or equal to the segment A'B'. In symbols, AB ≡ a'b'.
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Figure ??: Axiom III,1.
III,2.If a segment A'B' and a segment A''B'' are congruent to the same segment AB, then the segment A'B' is also congruent to the segment A''B''. Briefly, if two segments are congruent to a third one the are congruent to each other.
Sorry, this page requires a Java-compatible web browser.
Figure ??: Axiom III,2.
III,3.On a line let AB and BC be two segments which, except for B have no point in common. Furthermore, on the same or another line a', let A'B' and B'C' be two segments which, except for B' also have not points in common. In that case, if AB ≡ A'B' and BC ≡ B'C', then AC ≡ A'C'.
Sorry, this page requires a Java-compatible web browser.
Figure ??: Axiom III,3.

IV. Axiom of Parallels

V. Axioms of Continuity

References

  1. Hilbert, David. Foundations of Geometry. 2nd English addition. Translated by Leo Unger. Open Court. 1990.

More Information

  • McAdams, David E. Euclidean Geometry. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem.org. 6/19/2018. http://www.allmathwords.org/Euclidean%20Geometry.

Cite this article as:

McAdams, David E. Axioms of David Hilbert. 6/15/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/a/axiomshilbert.html.

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Revision History

6/15/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
5/1/2010: Initial version. (McAdams, David E.)

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