Hilbert's axioms
Encyclopedia
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
.
s: three primitive terms
and these six primitive relations:
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.
R. L. Moore proved that this axiom is redundant, in 1902.
atize 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 system
s. 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.
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry), as the foundation for a modern treatment of Euclidean geometry
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions from these...
. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski
Tarski's axioms
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory . Other modern axiomizations of Euclidean geometry are those by Hilbert and George...
and of George Birkhoff
Birkhoff's axioms
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the...
.
The axioms
Hilbert's axiom system is constructed with nine primitive notionPrimitive notion
In mathematics, logic, and formal systems, a primitive notion is an undefined concept. In particular, a primitive notion is not defined in terms of previously defined concepts, but is only motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory or...
s: three primitive terms
- pointPoint (geometry)In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
, straight line, planePlane (mathematics)In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...
,
and these six primitive relations:
- Betweenness, a ternary relation linking points;
- Containment, three binary relationBinary relationIn mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
s, 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 segmentLine segmentIn geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
s and one linking angleAngleIn geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...
s, 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
- 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.
- 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.
- 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.
- Any three points A, B, C of a plane α, which do not lie in the same straight line, completely determine that plane.
- 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.
- If two planes α, β have a point A in common, then they have at least a second point B in common.
- 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
- 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.
- 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.
- Of any three points situated on a straight line, there is always one and only one which lies between the other two.
- Pasch's AxiomPasch's axiomIn geometry, Pasch's axiom is a result of plane geometry used by Euclid, but yet which cannot be derived from Euclid's postulates. Its axiomatic role was discovered by Moritz Pasch.The axiom states that, in the plane,...
: 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
- 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
- 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 AB ≅ A′ B′. 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. - 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 AB ≅ A′B′ and AB ≅ A″B″, then A′B′ ≅ A″B″
- 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 AB ≅ A′B′ and BC ≅ B′C′, we have AC ≅ A′C′.
- 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) - 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″).
- 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
- 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.
- 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.
Hilbert's discarded axiom
Hilbert (1899) included a 21st axiom that read as follows:- II.4. Pasch's TheoremPasch's theoremIn geometry, Pasch's theorem, stated in 1882 by a German mathematician Moritz Pasch, is a result of plane geometry which cannot be derived from Euclid's postulates. It would now be considered as order theory, but the point it makes is in relation to the axiomatic method.The statement is as follows...
. 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.
R. L. Moore proved that this axiom is redundant, in 1902.
Application
These axioms axiomAxiom
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
atize Euclidean solid geometry
Solid geometry
In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane 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
Tarski's axioms
Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry, called "elementary," that is formulable in first-order logic with identity, and requiring no set theory . Other modern axiomizations of Euclidean geometry are those by Hilbert and George...
, do not constitute a first-order theory
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
because the axioms V.1–2 cannot be expressed in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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
Moritz Pasch
Moritz Pasch was a German mathematician specializing in the foundations of geometry. He completed his Ph.D. at the University of Breslau at only 22 years of age...
, Mario Pieri
Mario Pieri
Mario Pieri was an Italian mathematician who is known for his work on foundations of geometry.Pieri was born in Lucca, Italy, the son of Pellegrino Pieri and Ermina Luporini. Pellegrino was a lawyer. Pieri began his higher education at University of Bologna where he drew the attention of Salvatore...
, Oswald Veblen
Oswald Veblen
Oswald Veblen was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity. He proved the Jordan curve theorem in 1905.-Life:...
, Edward Vermilye Huntington
Edward Vermilye Huntington
Edward Vermilye Huntington was an American mathematician....
, Gilbert Robinson, and Henry George Forder. The value of the Grundlagen is its pioneering approach to metamathematical
Metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
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 system
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
s. 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.