Hilbert's axioms

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Hilbert's axioms are a set of 20 assumptions (originally 21), David Hilbert proposed in 1899 as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Tarski and of George Birkhoff.

Contents

[edit] The axioms

The undefined primitives are: point, line, plane. There are three primitive relations:

  • Betweenness, a ternary relation linking points;
  • Containment, three binary relations, one linking points and lines, one linking points and planes, and one linking 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 lines, using the relations of betweenness and containment.

All points, lines, and planes in the following axioms are distinct unless otherwise stated.

[edit] I. Incidence

I.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 and or with 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.

I.2: Any two distinct points of a straight line completely determine that line; that is, if AB = a and AC = a, where B ≠ C, then also BC = a.

I.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.

I.4: Any three points A, B, C of a plane α, which do not lie in the same straight line, completely determine that plane.

I.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.

I.6: If two planes α, β have a point A in common, then they have at least a second point B in common.

I.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.

[edit] II. Order

II.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.

II.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.

II.3: Of any three points situated on a straight line, there is always one and only one which lies between the other two.

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.

[edit] III. Parallels

III.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.

[edit] IV. Congruence

IV.1: If A, B are two points on a straight line a, and if Aˡ is a point upon the same or another straight line aˡ , then, upon a given side of Aˡ on the straight line aˡ , we can always find one and only one point Bˡ so that the segment AB (or BA) is congruent to the segment Aˡ Bˡ . We indicate this relation by writing AB ≅ Aˡ Bˡ. Every segment is congruent to itself; that is, we always have AB ≅ AB.

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.

IV.2: If a segment AB is congruent to the segment AˡBˡ and also to the segment AˡˡBˡˡ, then the segment AˡBˡ is congruent to the segment AˡˡBˡˡ; that is, if AB ≅ AˡB and AB ≅ AˡˡBˡˡ, then AˡBˡ ≅ AˡˡBˡˡ

IV.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 AˡBˡ and BˡCˡ be two segments of the same or of another straight line aˡ having, likewise, no point other than Bˡ in common. Then, if AB ≅ AˡBˡ and BC ≅ BˡCˡ, we have AC ≅ AˡCˡ.

IV.4: Let an angle (h, k) be given in the plane α and let a straight line aˡ be given in a plane αˡ. Suppose also that, in the plane α, a definite side of the straight line aˡ be assigned. Denote by hˡ a half-ray of the straight line aˡ emanating from a point Oˡ of this line. Then in the plane αˡ there is one and only one half-ray kˡ such that the angle (h, k), or (k, h), is congruent to the angle (hˡ, kˡ) and at the same time all interior points of the angle (hˡ, kˡ) lie upon the given side of aˡ. We express this relation by means of the notation ∠(h, k) ≅ (hˡ, kˡ)

Every angle is congruent to itself; that is, ∠(h, k) ≅ (h, k)

or

∠(h, k) ≅ (k, h)

IV.5: If the angle (h, k) is congruent to the angle (hˡ, kˡ) and to the angle (hˡˡ, kˡˡ), then the angle (hˡ, kˡ) is congruent to the angle (hˡˡ, kˡˡ); that is to say, if ∠(h, k) ≅ (hˡ, kˡ) and ∠(h, k) ≅ (hˡˡ, kˡˡ), then ∠(hˡ, kˡ) ≅ (hˡˡ, kˡˡ).

IV.6: If, in the two triangles ABC and AˡBˡCˡ the congruences AB ≅ AˡBˡ, AC ≅ AˡCˡ, ∠BAC ≅ ∠BˡAˡCˡ hold, then the congruences ∠ABC ≅ ∠AˡBˡCˡ and ∠ACB ≅ ∠AˡCˡBˡ also hold.

[edit] V. Continuity

V.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.

V.2: Line 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.

[edit] Discussion

[edit] Hilbert's discarded axiom

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

II.5: 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.

This may be understood a bit more simply as:

Pasch's Axiom : A line which intersects one edge of a triangle and misses the three vertices must intersect one of the other two edges.

R. L. Moore proved that this axiom is redundant, in 1902.

[edit] Application

These axioms axiomatize Euclidian 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 do not constitute a first-order theory because the axioms in group V cannot be expressed in first-order logic. Therefore Hilbert's axioms, unlike Tarski's, implicitly draw on set theory and so cannot be proved decidable or complete.

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.

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