Hilbert's axioms

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Some math characters may not display properly on the page (I, for example, can see congruence symbols in Firefox... but not in IE). They all appear correctly in the printable version.

A very brief background:

Mathematician David Hilbert (1862-1943) undertook a study of Euclid's geometry ... and essentially undertook the project of fixing it up (as we've seen, Euclid's geometry, while an amazing first attempt at a formal, axiom based geometry, has quite a few holes in it). He proposed 21 axioms (the list was later reduced to 20 when it was proven that one of the axioms was redundant) needed to establish a Euclidean - type geometry rigorously. These axioms form the basis for the modern treatment of geometry.

Hilbert biography

Hilbert's Axioms:

The undefined terms are: point, line, plane.

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.

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.

III. Congruence.

If you see empty boxes, they are probably congruence symbols. Refer to the printable version at the top of the page.

  • 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 ABA'C and ABA'D.
  • III.2: If CDAB and EFAB, then CDEF.
  • 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 ABA'B' and BCB'C' then ACA'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 ABA'B' , ACA'C' , and ∠BAC ≅ ∠B'A'C' , then ΔABC ≅ ΔA'B'C' .

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.

V. Continuity.

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


Incidence axioms:

The first group of axioms are the "incidence axioms" - "incident" in this context means "connected" or "touching", and to say a line is incident with a plane is to say the line lies in the plane. Incidence axioms are axioms about how points, lines, and planes connect to each other.

Kay's treatment:

So, when your text starts off talking about "incidence axioms," it's talking about the basic rules to connect points, lines and planes. Kay starts with five axioms, which correspond to Hilbert's in the following way:

  • Kay I-1 [Hilbert I-1, I-2]: Each two distinct points determine a line. [The "determine" is the "unique" part.]
  • Kay I-2 [Hilbert I -4, I-5]: Three noncollinear points determine a plane.
  • Kay I-3 [Hilbert I- 6]: If two points lie in a plane, then any line containing those two points lies in that plane.
  • Kay I-4 [Hilbert I-7]: If two distinct planes meet, their intersection is a line.
  • Kay I-5 [Hilbert I-8, I-3]: Space consists of at least four noncoplanar points, and contains three noncollinear points. Each plane is a set of points of at which three are noncollinear, and each line is a set of at least two distinct points.

College Geometry: A Discovery Approach (2e), David C. Kay

(You can see that all the essential items in Hilbert's eight are compressed into Kay's five)

The progression in Kay is to start with a small set of axioms, see what kinds of geometries can be developed, and then throw some more axioms into the mix and see where we can get from there. Much of the material in the next couple chapters is the familiar high school stuff again - what you want to keep an eye on is the order in which it gets built up, and how far you can on a set of axioms before you need to pick up another one!

Finite geometries:

Finally, we're used to our lines and planes containing an infinite number of points. Notice that's not part of the incidence axioms. We'll be starting out by looking at geometries using a finite (and rather small) number of points.