December 9

Comparison of Axiomatic Systems for Geometry:
Euclid, Hilbert, and Birkoff

Problems | Euclid's 28 Axioms | Hilbert's Axioms | Birkoff's 4 Axioms

Hilbert's Axioms

Hilbert's Axioms are divided into sections: Incidence Axioms, Betweenness Axioms, Congruence Axioms, Continuity Axiom, and Parallelism Axiom.

Incidence Axioms:

HI1. For every point P and for every point Q not equal to P, there exists a unique line incident with P and Q.

HI2. For every line there exists at least two distinct points that are incident with .

HI3. There exist three distinct points with the property that no line is incident with all three of them.

Betweenness Axioms - the symbol "*" means "lies between"

HB1. If A * B * C, then A, B, and C are three distinct points all lying on the same line,
and C * B * A.

HB3. If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.

HB4. For every line and for any three points A, B, and C not lying on :

(i) If A and B are on the same side of and
B and C are on the same side of , then A and C are on the same side of .

(ii) If A and B are on opposite sides of and
B and C are on opposite sides of , then A and C are on the same side of .

Congruence Axioms

HC5. If < A is congruent to < B and < A is congruent to < C, then < B is congruent to < C.
Moreover, every angle is congruent to itself.

HC6. If two sides and the included angle of one triangle are congruent respectively to two
sides and the included angle of another triangle, then the two triangles are congruent.

Continuity Axiom

Parallelism Axiom

For every line and every point P not lying on there is at most one line m through P such that m is parallel to .

Euclidean and Non-Euclidean Geometries: Development and History
by M. J. Greenberg
pp. 469 - 471

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Birkhoff's 4 Postulates

B1.Postulate of line measure: The points, A, B, ... of any line can be put into (1,1) correspondence with the real numbers x
so that | xB - xA | = d(A, B) for all points A,B.

Modern Geometries
by J. Smart
p. 417

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