Hellenica World

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Hilbert's axioms are a set of 20 (originally 21) assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry), as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.

The axioms

Hilbert's axiom system is constructed with nine primitive notions: three primitive terms

point,  straight line,  plane,

and these six primitive relations:

• Betweenness, a ternary relation linking points;
• Containment, three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes;
• Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅.

Note that line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment. All points, straight lines, and planes in the following axioms are distinct unless otherwise stated.

I. Combination

1. Two distinct points A and B always completely determine a straight line a. We write AB = a or BA = a. Instead of “determine,” we may also employ other forms of expression; for example, we may say “A lies upon a”, “A is a point of a”, “a goes through A and through B”, “a joins A to B”, etc. If A lies upon a and at the same time upon another straight line b, we make use also of the expression: “The straight lines a and b have the point A in common,” etc.
2. Any two distinct points of a straight line completely determine that line; that is, if AB = a and AC = a, where BC, then also BC = a.
3. Three points A, B, C not situated in the same straight line always completely determine a plane α. We write ABC = α. We employ also the expressions: “A, B, C, lie in α”; “A, B, C are points of α”, etc.
4. Any three points A, B, C of a plane α, which do not lie in the same straight line, completely determine that plane.
5. If two points A, B of a straight line a lie in a plane α, then every point of a lies in α. In this case we say: “The straight line a lies in the plane α,” etc.
6. If two planes α, β have a point A in common, then they have at least a second point B in common.
7. Upon every straight 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. Order

1. If a point B is between points A and C, B is also between C and A, and there exists a line containing the points A,B,C.
2. If A and C are two points of a straight line, then there 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.
3. Of any three points situated on a straight line, there is always one and only one which lies between the other two.
1. Pasch's Axiom: Let A, B, C be three points not lying in the same straight line and let a be a straight line lying in the plane ABC and not passing through any of the points A, B, C. Then, if the straight line a passes through a point of the segment AB, it will also pass through either a point of the segment BC or a point of the segment AC.

III. Parallels

1. In a plane α there can be drawn through any point A, lying outside of 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. Congruence

1. If A, B are two points on a straight line a, and if A′ is a point upon the same or another straight line a′ , then, upon a given side of A′ on the straight 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′ . We indicate this relation by writing ABA′ B′. Every segment is congruent to itself; that is, we always have ABAB.
We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in one and only one way.
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″; that is, if ABA′B′ and ABA″B″, then A′B′A″B″
3. Let AB and BC be two segments of a straight line a which have no points in common aside from the point B, and, furthermore, let A′B′ and B′C′ be two segments of the same or of another straight line a′ having, likewise, no point other than B′ in common. Then, if ABA′B′ and BCB′C′, we have ACA′C′.
4. Let an angle (h, k) be given in the plane α and let a straight line a′ be given in a plane α′. Suppose also that, in the plane α′, a definite side of the straight line a′ be assigned. Denote by h′ a half-ray of the straight line a′ emanating from a point O′ of this line. Then in the plane α′ there is one and only one half-ray k′ such that the angle (h, k), or (k, h), is congruent to the angle (h′, k′) and at the same time all interior points of the angle (h′, k′) lie upon the given side of a′. We express this relation by means of the notation ∠(h, k) ≅ (h′, k′)
Every angle is congruent to itself; that is, ∠(h, k) ≅ (h, k)
or
∠(h, k) ≅ (k, h)
5. If the angle (h, k) is congruent to the angle (h′, k′) and to the angle (h″, k″), then the angle (h′, k′) is congruent to the angle (h″, k″); that is to say, if ∠(h, k) ≅ (h′, k′) and ∠(h, k) ≅ (h″, k″), then ∠(h′, k′) ≅ (h″, k″).
6. If, in the two triangles ABC and A′B′C′ the congruences AB ≅ A′B′, AC ≅ A′C′, ∠BAC ≅ ∠B′A′C′ hold, then the congruences ∠ABC ≅ ∠A′B′C′ and ∠ACB ≅ ∠A′C′B′ also hold.

V. Continuity

1. Axiom of Archimedes. Let A1 be any point upon a straight line between the 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, A3A4, . . . be 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.
2. Axiom of completeness. To a system of points, straight lines, and planes, it is impossible to add other elements in such a manner that the system thus generalized shall form a new geometry obeying all of the five groups of axioms. In other words, the elements of geometry form a system which is not susceptible of extension, if we regard the five groups of axioms as valid.

Hilbert (1899) included a 21st axiom that read as follows:

II.4. Pasch's Theorem. Any four points A, B, C, D of a straight line can always be so arranged 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.

E.H. Moore and R.L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in the Transactions of the American Mathematical Society in 1902.
Editions and translations of Grundlagen der Geometrie

The original monograph, based on his own lectures, was organized and written by Hilbert for a memorial address given in 1899. This was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime. In the Preface of this edition Hilbert wrote:

"The present Seventh Edition of my book Foundations of Geometry brings considerable improvements and additions to the previous edition, partly from my subsequent lectures on this subject and partly from improvements made in the meantime by other writers. The main text of the book has been revised accordingly."

New editions followed the 7th, but the main text was essentially not revised. The modifications in these editions occur in the appendices and in supplements. The changes in the text were large when compared to the original and a new English translation was commissioned by Open Court Publishers, who had published the Townsend translation. So, the 2nd English Edition was translated by Leo Unger from the 10th German edition in 1971. This translation incorporates several revisions and enlargements of the later German editions by Paul Bernays.

The Unger translation differs from the Townsend translation with respect to the axioms in the following ways:

• Old axiom II.4 (Pasch's Theorem) is renamed as Theorem 4 and moved.
• Old axiom II.5 (Pasch's Axiom) is renumbered as II.4.
• V.2, the Axiom of Completeness, has been replaced by:
The Axiom of Line Completeness. An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follows from Axioms I-III, and from V.1 is impossible.
• The old axiom V.2 is now Theorem 32.

The last two modifications are due to P. Bernays.

Application

These axioms axiomatize Euclidean solid geometry. Removing four axioms mentioning "plane" in an essential way, namely I.3–6, omitting the last clause of I.7, and modifying III.1 to omit mention of planes, yields an axiomatization of Euclidean plane geometry.

Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V.1–2 cannot be expressed in first-order logic.

The value of Hilbert's Grundlagen was more methodological than substantive or pedagogical. Other major contributions to the axiomatics of geometry were those of Moritz Pasch, Mario Pieri, Oswald Veblen, Edward Vermilye Huntington, Gilbert Robinson, and Henry George Forder. The value of the Grundlagen is its pioneering approach to metamathematical questions, including the use of models to prove axioms independent; and the need to prove the consistency and completeness of an axiom system.

Mathematics in the twentieth century evolved into a network of axiomatic formal systems. This was, in considerable part, influenced by the example Hilbert set in the Grundlagen. A 2003 effort (Meikle and Fleuriot) to formalize the Grundlagen with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions.
References

Howard Eves, 1997 (1958). Foundations and Fundamental Concepts of Mathematics. Dover. Chpt. 4.2 covers the Hilbert axioms for plane geometry.
Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton University Press.
David Hilbert, 1980 (1899). The Foundations of Geometry, 2nd ed. Chicago: Open Court.
Laura I. Meikle and Jacques D. Fleuriot (2003), Formalizing Hilbert's Grundlagen in Isabelle/Isar, Theorem Proving in Higher Order Logics, Lecture Notes in Computer Science, Volume 2758/2003, 319-334, doi:10.1007/10930755_21