Hilbert's Axiomatic System 📂Geometry

Hilbert's Axiomatic System

Overview

Hilbert accepted basic terms such as point, line, plane, “on”, “between”, and congruence without defining them, and left 16 statements grouped into five categories in order to establish an axiom system for geometry. In particular, the axiom of continuity clarified the precise relationship between Euclidean geometry and arithmetic systems.

Hilbert’s statements about Euclidean geometry addressed most of the problems that the Elements of Euclid had, and made the mathematics of Hilbert’s time acceptable to mathematical intuition. The Hilbert axiom system is said to have had a level of rigor that later became a standard in mathematics.

Axioms ¹

I. 연결 공리

For any two points $A$ and $B$, there exists a line that contains both $A$ and $B$.
For any two points $A$ and $B$, the line containing $A$ and $B$ is unique.
There exist at least two points on every line. There exists at least one point not lying on a given line.
For any three points $A$, $B$, $C$ not all on the same line, there exists a plane $\alpha$ containing $A$, $B$, and $C$. Every plane contains at least one point.

II. 순서 공리

If point $B$ lies between $A$ and $C$, then $A$, $B$, and $C$ are distinct points on the same line and $B$ also lies between $C$ and $A$.
For any distinct points $A$ and $C$, there exists at least one point $B$ on line $\overleftrightarrow{AC}$ such that $C$ lies between $A$ and $B$.
If $A$, $B$, $C$ are three points on the same line, then the point lying between the other two is unique.
Let $A$, $B$, $C$ be three noncollinear points, and let $l$ be a line in the plane determined by $A$, $B$, $C$ that does not meet any of $A$, $B$, $C$. If $l$ passes through a point of segment $\overline{AB}$, then $l$ also passes through a point of segment $\overline{AC}$ or a point of $\overline{BC}$.

III. 합동 공리

If $A$ and $B$ are two points on line $a$, and $A '$ is a point on the same or a different line $a '$, then one can always find a point $B '$ on $\alpha '$ such that segments $\overline{AB}$ and $\overline{A ' B '}$ are congruent.
If segments $\overline{A ' B '}$ and $\overline{A '' B ''}$ are congruent to the same segment $\overline{AB}$, then $\overline{A ' B '}$ and $\overline{A '' B ''}$ are congruent to each other.
On line $a$, let segments $\overline{AB}$ and $\overline{BC}$ have no point in common except point $B$. On the same or a different line $a '$, let segments $\overline{A ' B '}$ and $\overline{B ' C '}$ have no point in common except point $B '$. If $\overline{AB}$ is congruent to $\overline{A ' B '}$ and $\overline{BC}$ is congruent to $\overline{B ' C '}$, then $\overline{AC}$ and $\overline{A ' C '}$ are congruent.
If $\angle{ABC}$ is an angle and $\overrightarrow{B ' C '}$ is a ray, then there exists a ray $\overrightarrow{B ' A '}$ on one side of line $\overrightarrow{B ' C '}$ such that $\angle{A ' B ' C '}$ is congruent to $\angle{ABC}$.
If two triangles $\triangle{ABC}$ and $\triangle{A ' B ' C '}$ satisfy $\overline{AB} \cong \overline{A ' B '}$, $\overline{AC} \cong \overline{A ' C '}$, $\angle{BAC} \cong \angle{B ' A ' C '}$, then $\angle{ABC}$ and $\angle{A ' B ' C '}$ are congruent.

IV. 평행 공리

Let $A$ be a point not on line $a$. Then, in the plane determined by $a$ and $A$, there is at most one line through $A$ that does not meet $a$.

V. 연속 공리

아르키메데스의 공리: If $\overline{AB}$ and $\overline{CD}$ are segments, then there exists a natural number $n$ such that laying down $n$ copies of $\overline{CD}$ from $A$ along $\overrightarrow{AB}$ in a row will surpass $B$.
직선 완비성의 공리: It is impossible to extend the set of points of an existing line—while preserving the original order and congruence relations—to obtain a line that satisfies Axioms I–III and V-1 and properly contains the original line.

Remarks

It is immediately apparent that, compared with the entirely non-symbolic Euclidean axioms, Hilbert’s formulation is more modern and much easier to read.

Notably, the parallel axiom introduces an explicit restriction to the “plane,” which is not mentioned in the original Euclidean formulation.
The Archimedean axiom pleasantly corresponds to the Archimedean principle in analysis expressed in the language of geometry.
The axiom of line completeness means that a line is complete in itself: although a line already contains sufficiently many points, it is impossible to add more points to form a strictly larger line that preserves the required order and congruence relations.