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Hilbert's axioms

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Basis for Euclidean geometry

Hilbert's axioms are a set of 20 assumptions proposed byDavid Hilbert in 1899 in his bookGrundlagen der Geometrie^[1]^[2]^[3]^[4] (tr.The Foundations of Geometry) as the foundation for a modern treatment ofEuclidean geometry. Other well-known modernaxiomatizations of Euclidean geometry are those ofAlfred Tarski and ofGeorge Birkhoff.

The axioms

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Hilbert'saxiom system is constructed with sixprimitive notions: three primitive terms:^[5]

point (represented by Latin capital letters, $A,B,C,\dots$ );
line (represented by Latin minuscules, $l,m,n,\dots$ );
plane (represented by Greek minuscules, $\pi ,\rho ,\sigma ,\dots$ );

and three primitive relations:

$A\in l$ or $A\in \pi$ or $l\in \pi$ —incidence (a point lies on a line, a point lies in a plane, or a line lies in a plane)
$A*B*C$ —betweenness, meaning that $B {\displaystyle B}$ is between $A {\displaystyle A}$ and $C {\displaystyle C}$
${\overline {AB}}\cong {\overline {CD}}$ —segment congruence, stating that the segments $A B {\displaystyle AB}$ and $C D {\displaystyle CD}$ are equal in length
$\angle ABC\cong \angle DEF$ —angle congruence, stating that the angles $A B C {\displaystyle ABC}$ and $D E F {\displaystyle DEF}$ are equal in measure

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.

IfA,B are two points on a linea, and ifA′ is a point upon the same or another linea′, then, upon a given side ofA′ on the straight linea′, we can always find a pointB′ so that the segmentAB is congruent to the segmentA′B′. We indicate this relation by writingAB ≅A′B′. Every segment is congruent to itself; that is, we always haveAB ≅AB.
We can state the above axiom briefly by saying that every segment can belaid off upon a given side of a given point of a given straight line in at least one way.
${\begin{aligned}(C_{1})&\quad \forall A,B,A',B'\,\exists C'\,({\text{Ray}}(A',B',C')\wedge {\overline {AB}}\cong {\overline {A'C'}})\end{aligned}}$
If a segmentAB is congruent to the segmentA′B′ and also to the segmentA″B″, then the segmentA′B′ is congruent to the segmentA″B″; that is, ifAB ≅A′B′ andAB ≅A″B″, thenA′B′ ≅A″B″.
${\begin{aligned}(C_{2})&\quad \forall A,B\,({\overline {AB}}\cong {\overline {AB}})\end{aligned}}$ (reflexive)
${\begin{aligned}(C_{2})&\quad \forall A,B,C,D\,(({\overline {AB}}\cong {\overline {CD}})\rightarrow ({\overline {CD}}\cong {\overline {AB}}))\end{aligned}}$ (symmetric)
${\begin{aligned}(C_{2})&\quad \forall A,B,C,D,E,F\,(({\overline {AB}}\cong {\overline {CD}}\wedge {\overline {CD}}\cong {\overline {EF}})\rightarrow ({\overline {AB}}\cong {\overline {EF}}))\end{aligned}}$ (transitive)
LetAB andBC be two segments of a linea which have no points in common aside from the pointB, and, furthermore, letA′B′ andB′C′ be two segments of the same or of another linea′ having, likewise, no point other thanB′ in common. Then, ifAB ≅A′B′ andBC ≅B′C′, we haveAC ≅A′C′.
${\begin{aligned}(C_{3})&\quad \forall A,B,C,A',B',C'\,(({\overline {AB}}\cong {\overline {A'B'}}\wedge {\overline {BC}}\cong {\overline {B'C'}})\rightarrow {\overline {AC}}\cong {\overline {A'C'}})\end{aligned}}$
Let an angle∠ (h,k) be given in the planeα and let a linea′ be given in a planeα′. Suppose also that, in the planeα′, a definite side of the straight linea′ be assigned. Denote byh′ a ray of the straight linea′ emanating from a pointO′ of this line. Then in the planeα′ there is one and only one rayk′ 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 ofa′. We express this relation by means of the notation∠ (h,k) ≅ ∠ (h′,k′).
${\begin{aligned}(C_{4})&\quad \forall A,B,C,A',B'\,\exists !\,C'\,(\angle ABC\cong \angle A'B'C')\end{aligned}}$
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″).
${\begin{aligned}(C_{5})&\quad \forall A,B,C\,(\angle ABC\cong \angle ABC)\end{aligned}}$
${\begin{aligned}(C_{5})&\quad \forall A,B,C,D,E,F\,((\angle ABC\cong \angle DEF)\rightarrow (\angle DEF\cong \angle ABC))\end{aligned}}$
${\begin{aligned}(C_{5})&\quad \forall A,B,C,D,E,F,G,H,I\,((\angle ABC\cong \angle DEF\wedge \angle DEF\cong \angle GHI)\rightarrow \angle ABC\cong \angle GHI)\end{aligned}}$
If, in the two trianglesABC andA′B′C′ the congruencesAB ≅A′B′,AC ≅A′C′,∠BAC ≅ ∠B′A′C′ hold, then the congruence∠ABC ≅ ∠A′B′C′ holds (and, by a change of notation, it follows that∠ACB ≅ ∠A′C′B′ also holds).
${\begin{aligned}(C_{6})&\quad \forall A,B,C,A',B',C'\,(({\overline {AB}}\cong {\overline {A'B'}}\wedge {\overline {AC}}\cong {\overline {A'C'}}\wedge \angle BAC\cong \angle B'A'C')\rightarrow (\angle {ACB}\cong \angle {A'C'B'}))\end{aligned}}$

IV. Parallels

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Playfair's axiom:^[8] Leta be any line andA a point not on it. Then there is at most one line in the plane, determined bya andA, that passes throughA and does not intersecta.

${\begin{aligned}(P)\quad \forall A\,\forall l\ {\bigl (}A\notin l\rightarrow \exists !m\ &(A\in m\wedge \neg \exists P\,(P\in l\wedge P\in m)){\bigr )}.\end{aligned}}$

V. Continuity

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Axiom of Archimedes: IfAB andCD are any segments then there exists a numbern such thatn segmentsCD constructed contiguously fromA, along the ray fromA throughB, will pass beyond the pointB.
Axiom of line completeness: An extension (An extended line from a line that already exists, usually used in geometry) 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.

${\begin{aligned}(Ct)\quad &\forall l\ \forall X\subseteq \{P:P\in l\}\ \forall Y\subseteq \{P:P\in l\}\ {\Bigl [}\,X\neq \varnothing \wedge Y\neq \varnothing \wedge X\cup Y=\{P:P\in l\}\\&\qquad \wedge X\cap Y=\varnothing \wedge (\forall x\in X\ \forall y\in Y\ \neg \exists z\,(z\in l\wedge y*z*x))\\&\qquad \longrightarrow \exists c\,(c\in l\wedge \forall x\in X\ \forall y\in Y\ (x*c*y\vee x=c\vee y=c))\,{\Bigr ]}\end{aligned}}$

Hilbert's discarded axiom

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

II.4. Any four pointsA,B,C,D of a line can always be labeled so thatB shall lie betweenA andC and also betweenA andD, and, furthermore, thatC shall lie betweenA andD and also betweenB andD.

This statement is also known asPasch's theorem.

E. H. Moore andR. L. Moore independently proved that this axiom is redundant, and the former published this result in an article appearing in theTransactions of the American Mathematical Society in 1902.^[9]

Before this,Pasch's axiom, now listed as II.4, was numbered II.5.

Editions and translations ofGrundlagen der Geometrie

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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 bookFoundations 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 is renamed as Theorem 5 and moved.
Old axiom II.5 (Pasch's Axiom) is renumbered as II.4.
V.2, the Axiom of Line Completeness, replaced:

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.

The old axiom V.2 is now Theorem 32.

The last two modifications are due to P. Bernays.

Other changes of note are:

The termstraight line used by Townsend has been replaced byline throughout.
TheAxioms of Incidence were calledAxioms of Connection by Townsend.

Application

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These axiomsaxiomatize Euclideansolid geometry. Removing five axioms mentioning "plane" in an essential way, namely I.4–8, and modifying III.4 and IV.1 to omit mention of planes, yields an axiomatization ofEuclidean plane geometry.

Hilbert's axioms, unlikeTarski's axioms, do not constitute afirst-order theory because the axioms V.1–2 cannot be expressed infirst-order logic.

The value of Hilbert'sGrundlagen was more methodological than substantive or pedagogical. Other major contributions to the axiomatics of geometry were those ofMoritz Pasch,Mario Pieri,Oswald Veblen,Edward Vermilye Huntington,Gilbert Robinson, andHenry George Forder. The value of theGrundlagen is its pioneering approach tometamathematical 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 axiomaticformal systems. This was, in considerable part, influenced by the example Hilbert set in theGrundlagen. A 2003 effort (Meikle and Fleuriot) to formalize theGrundlagen 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.^[10]

Notes

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^Sommer, Julius (1900)."Review: Grundlagen der Geometrie, Teubner, 1899"(PDF).Bull. Amer. Math. Soc.6 (7):287–299.doi:10.1090/s0002-9904-1900-00719-1.
^Poincaré, Henri (1903)."Poincaré's review of Hilbert's "Foundations of Geometry", translated by E. V. Huntington"(PDF).Bull. Amer. Math. Soc.10:1–23.doi:10.1090/S0002-9904-1903-01061-1.
^Schweitzer, Arthur Richard (1909)."Review:Grundlagen der Geometrie, Third edition, Teubner, 1909"(PDF).Bull. Amer. Math. Soc.15 (10):510–511.doi:10.1090/s0002-9904-1909-01814-2.
^Gronwall, T. H. (1919)."Review:Grundlagen der Geometrie, Fourth edition, Teubner, 1913"(PDF).Bull. Amer. Math. Soc.20 (6):325–326.doi:10.1090/S0002-9904-1914-02492-9.
^These axioms and their numbering are taken from the Unger translation (into English) of the 10th edition ofGrundlagen der Geometrie.
^In the Townsend edition this statement differs in that it also includes the existence of at least one pointD withC betweenA andD, which became a theorem in a later edition.
^The existence part ("there is at least one") is a theorem.
^This is Hilbert's terminology. This statement is more familiarly known asPlayfair's axiom.
^Moore, E. H. (1902),"On the projective axioms of geometry"(PDF),Transactions of the American Mathematical Society,3 (1):142–158,doi:10.2307/1986321,JSTOR 1986321
^On page 334:"By formalizing theGrundlagen in Isabelle/Isar we showed that Hilbert's work glossed over subtle points of reasoning and relied heavily, in some cases, on diagrams which allowed implicit assumptions to be made. For this reason it can be argued that Hilbert interleaved his axioms with geometric intuition in order to prove many of his theorems."

References

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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 Archived 2016-03-04 at theWayback Machine, Theorem Proving in Higher Order Logics, Lecture Notes in Computer Science, Volume 2758/2003, 319-334,doi:10.1007/10930755_21

External links

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"Hilbert system of axioms",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
"Hilbert's Axioms" at the UMBC Math Department
"Hilbert's Axioms" at Mathworld
Foundations of Geometry public domain audiobook atLibriVox

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