Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
The axioms Hilbert's axiom system is constructed with six primitive notions: three primitive terms:
point (represented by Latin capital letters, A , B , C , … {\displaystyle A,B,C,\dots } ); line (represented by Latin minuscules, l , m , n , … {\displaystyle l,m,n,\dots } ); plane (represented by Greek minuscules, π , ρ , σ , … {\displaystyle \pi ,\rho ,\sigma ,\dots } ); and three primitive relations:
A ∈ l {\displaystyle A\in l} or A ∈ π {\displaystyle A\in \pi } or l ∈ π {\displaystyle 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 {\displaystyle A*B*C} — betweenness, meaning that B {\displaystyle B} is between A {\displaystyle A} and C {\displaystyle C}
A B ¯ ≅ C D ¯ {\displaystyle {\overline {AB}}\cong {\overline {CD}}} — segment congruence, stating that the segments A B {\displaystyle AB} and C D {\displaystyle CD} are equal in length
∠ A B C ≅ ∠ D E F {\displaystyle \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.
I. Incidence
For every two points A and B there exists a line a that contains them both. We write AB = a or BA = a. Instead of "contains", we may also employ other forms of expression; for example, we may say "A lies upon a", "A is a point of a", "a goes through A and through B", "a joins A to B", etc. If A lies upon a and at the same time upon another line b, we make use also of the expression: "The lines a and b have the point A in common", etc. For every two points there exists no more than one line that contains them both; consequently, if AB = a and AC = a, where B ≠ C, then also BC = a.
( I 1 ) ∀ A ∀ B ( A ≠ B → ∃ ! l ( A ∈ l ∧ B ∈ l ) ) {\displaystyle {\begin{aligned}(I_{1})&\quad \forall A\,\forall B\,(A\neq B\rightarrow \exists !l\ (A\in l\wedge B\in l))\\[4pt]\end{aligned}}}
There exist at least two points on a line.
( I 2 ) ∀ l ∃ A ∃ B ( A ≠ B ∧ A ∈ l ∧ B ∈ l ) {\displaystyle {\begin{aligned}(I_{2})&\quad \forall l\,\exists A\,\exists B\,(A\neq B\wedge A\in l\wedge B\in l)\\[4pt]\end{aligned}}}
There exist at least three points that do not lie on the same line.
( I 3 ) ∃ A ∃ B ∃ C ¬ ∃ l ( A ∈ l ∧ B ∈ l ∧ C ∈ l ) {\displaystyle {\begin{aligned}(I_{3})&\quad \exists A\,\exists B\,\exists C\ \neg \exists l\,(A\in l\wedge B\in l\wedge C\in l)\\[4pt]\end{aligned}}}
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