Math 2.0

Geometry Definition

Hilbert's axioms

Builds on LinePlaneBetweennessPasch's axiomPlayfair's axiom

Definition

The primitive objects are points, lines and planes. The primitive relations are incidence, betweenness of three points, and congruence ≅\cong of segments and of angles. Hilbert's axioms are the following five groups of axioms on these notions.

  1. Incidence: two distinct points lie on exactly one line. Three non-collinear points lie on exactly one plane. A line with two points in a plane lies in the plane. Two planes with a common point have a second common point. Each line has at least two points, each plane has three non-collinear points, and there are four points not in one plane.
  2. Order: the betweenness relation is symmetric in AA and CC and holds only for three distinct collinear points. For distinct AA and CC some point BB has CC between AA and BB. Of three collinear points, at most one lies between the other two. Pasch's axiom holds.
  3. Congruence: a segment can be laid off on a given ray in a unique way, and congruence of segments is transitive. Congruent adjacent segments add up to congruent segments. An angle can be laid off on a given side of a given ray in a unique way. If two triangles have two sides and the included angle congruent, then the other angles are congruent too.
  4. Parallels: Playfair's axiom holds.
  5. Continuity: the axiom of Archimedes holds for segments. The points of a line admit no extension that keeps all the other axioms (line completeness).

Remarks

Example: ℝ3\R^3 with its usual points, lines, planes, betweenness and congruence by isometries is a model. For instance, Pasch's axiom holds there because a line in a plane is the zero set of an affine function. Up to isomorphism, it is the only model, so the system is categorical. Non-example: the space ℚ3\Q^3 with the same notions fails the first congruence axiom. The segment from (0,0,0)(0,0,0) to (1,1,0)(1,1,0) has length 2\sqrt 2, so it cannot be laid off on the positive xx-axis from the origin, since (2,0,0)∉ℚ3(\sqrt 2, 0, 0) \notin \Q^3. The plane version keeps only the axioms of groups 1 to 5 that speak of one plane.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
Absolute geometry, concept, no evidence levelAbsolute geometryconceptBetweenness, concept, no evidence levelBetweennessconceptLine, concept, no evidence levelLineconceptPlane, concept, no evidence levelPlaneconceptTriangle, concept, no evidence levelTriangleconceptPlayfair's axiom, statement, no evidence levelPlayfair's axiomstatementPasch's axiom, statement, no evidence levelPasch's axiomstatementHilbert's axioms, definition, no evidence levelHilbert's axiomsdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Absolute geometry concept
  2. Betweenness concept
  3. Line concept
  4. Plane concept
  5. Triangle concept
  6. Playfair's axiom statement
  7. Pasch's axiom statement
  8. Hilbert's axioms definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%27s_axioms
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Kinddefinition
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MSC51M05, 51F05, 03B30
DefinesHilbert's axioms
UsesLine, Plane, Betweenness, Pasch's axiom, Playfair's axiom

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