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 axiomizations of Euclidean geometry are those of Tarski and of George Birkhoff.

1 The Axioms
2 Discussion
3 References
4 External links

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The Axioms

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

Betweenness, linking points;
Contain, linking points and lines, points and planes, or lines and planes;
Congruent, linking angles or triangles, and denoted by an infix ≅.

All points, lines, and planes are distinct unless otherwise stated.

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I. Incidence.

I.1: Given any two points, there exists a line containing both of them.

I.2: Given any two points, there exists no more than one line containing both points. I.e., the line described in I.1 is unique.

I.3: A line contains at least two points, and given any line, there exists at least one point not on it.

I.4: Given any three points not contained in one line, there exists a plane containing all three points. Every plane contains at least one point.

I.5: Given any three points not contained in one line, there exists only one plane containing all three points.

I.6: If two points contained in line m lie in some plane α, then α contains every point in m.

I.7: If the planes α and β both contain the point A, then α and β both contain at least one other point.

I.8: There exist at least four points not all contained in the same plane.

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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: Given two points A and C, there exists a point B on the line AC such that C lies between A and B.

II.3: Given any three points contained in one line, one and only one of the three points is between the other two.

II.4: Axiom of Pasch. Given three points A, B, C not contained in one line, and given a line m contained in the plane ABC but not containing any of A, B, C: if m contains a point on the segment AB, then m also contains a point on the segment AC or on the segment BC.

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III. Congruence.

III.1: Given two points A,B, and a point A' on line m, there exist two and only two points C and D, such that A' is between C and D, and AB ≅ A'C and AB ≅ A'D.

III.2: If CD ≅ AB and EF ≅ AB, then CD ≅ EF.

III.3: Let line m include the segments AB and BC whose only common point is B, and let line m or m' include the segments A'B' and B'C' whose only common point is B' . If AB ≅ A'B' and BC ≅ B'C' then AC ≅ A'C' .

III.4: Given the angle ∠ABC and ray B'C' , there exist two and only two rays, B'D and B'E,such that ∠DB'C' ≅ ∠ABC and ∠EB'C' ≅ ∠ABC.

Corollary: Every angle is congruent to itself.

III.5: Given two triangles ΔABC and ΔA'B'C' such that AB ≅ A'B' , AC ≅ A'C' , and ∠BAC ≅ ∠B'A'C' , then ΔABC ≅ ΔA'B'C' .

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IV. Parallels.

IV.1: Playfair's postulate. Given a line m, a point A not on m, and a plane containing both m and A: in that plane, there is at most one line containing A and not containing any point on m.

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V. Continuity.

V.1: Axiom of Archimedes. Given the line segment CD and the ray AB, there exist n points A₁,...,A_n on AB, such that A_jA_j+1 ≅ CD, 1≤j<n. Moreover, B is between A₁ and A_n.

V.2: Line completeness. Adding points to a line results in an object that violates one or more of the following axioms: I, II, III.1-2, V.1.

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Discussion

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

"Given any four points on a line, it is always possible to assign them the names A, B, C, and D, such that B is between A and C and A and D. Likewise, C will be between A and D and also between B and D."

E. H. Moore proved this axiom redundant in 1902.

These axioms axiomatize Euclidian solid geometry. Removing the five axioms mentioning "plane" in an essential way, namely I.4-8 and modifying IV.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 universe contains not one but three sorts of individuals: points, lines, and planes. If lines and planes are seen as sets of points, then 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. Moreover, Hilbert was far from the only one in his day to make major contributions to the axiomatics of geometry: Moritz Pasch, Mario Pieri, Oswald Veblen, Edward V. Huntington, Gilbert Robinson, and Henry George Forder all come to mind. Rather, the value of the Grundlagen is its pioneering approach to metamathematical questions, including the use of models to prove axioms independent, the need to prove the consistency and completeness of an axiom system, and so on. Mathematics evolved into a network of axiomatic formal systems in considerable part because of the example Hilbert set in the Grundlagen.

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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.

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