J. Math. Tokushima Univ.
Vol. 57 (2023), 93−100

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System of Axioms of Euclidean Geometry
By
Yoshifumi Ito
Professor Emeritus, Tokushima University
209-15 Kamifukuman Hachiman-cho
Tokushima 770-8073, JAPANe-mail address : itoyoshifumi@fd5.so-net.ne.jp(Received September 30, 2023)
Abstract
In this paper, we define Euclidean geometry and prove the existence
theorem of Euclidean geometry by using the axiomatic method.
Thereby we give the system of Axioms of Euclidean Geometry andprove its consistency.
Thus we give the complete solution of the problem of the foundationof Euclidean geometry.
2000 Mathematics Subject Classification. Primary, 51M05.

Introduction
In this paper, we give the true expression of Euclid’s system of axioms.
Thereby we deﬁne Euclidean geometry and prove the existence theorem of
Euclidean geometry.
Euclidean geometry was originated by Euclid’s “Principle”. In Euclid’s
system of axioms, the concepts of the point, the line and the plane were notdeﬁned clearly.
Therefore there was the problem of its truth. Hilbert gave the regularizationof Euclidean geometry by using Hilbert’s system of axioms in his “foundationof geometry”.

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Yoshifumi Ito

Before Cantor’s set theory, any one does not know the mathematical concepts of the point, the line and the plane.
Euclid’s system of axioms and Hilbert’s system of axioms do not give theconcrete plane geometry. They are only the abstract theory.
Namely, in these systems of axioms, they could not deﬁne concretely theconcepts of the point, the line and the plane.
By using the Cantor’s set theory or the exact set theory, we can deﬁne theEuclidean plane, and the line and the point there.
Therefore, by deﬁning the Euclidean plane deﬁned by using Weyl’s systemof axioms, we can complete the Euclid’s “Principle”.
At last, we express my heartfelt thanks to my wife Mutuko for her cooperation of making this ﬁle of the manuscript.

1 Proof of the consistency of the system of axioms of Euclidean geometry
In this section, we deﬁne Euclidean geometry and prove its existence theorem by constructing its model. Thereby we can prove the consistency of thesystem of axioms of Euclidean geometry. As for these facts, we refer to Ito [8],“Foundation of Linear algebra”, Chapter 16.
Definition 1.1(Euclidean geometry) Euclidean geometry is the geometry of the ﬁgures constructed by points and lines in the plane, which satisﬁes
the axioms (1)∼(5) in the following:(1)

We can draw the line from an arbitrary point to another arbitrary point.

(2)

We can extend the ﬁnite line (the line segment) to a straight line continuously.

(3)

We can draw a circle with an arbitrary point and a distance (the radius).

(4)

All right angles are mutually equal.

(5)

If one straight line intersects two straight lines and the sum of the innerangles on the same side is smaller than the 2-times right angles, the twostraight lines intersect on the side where the sum of the inner angles issmaller than the 2-times right angles when these two straight lines arebelonged inﬁnitely.

Remark 1.1 In Deﬁnition 1.1, Euclidean geometry is the geometry of theplanar ﬁgures drawn by using the ruler and the compass on the plane.
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Next, we prove the existence theorem of Euclidean geometry. For that
purpose, we prove this by constructing a model of Euclidean geometry.
Thereby we can prove the consistency of the system of axioms of Euclideangeometry.
Definition 1.2 (Plane) We assume that the set of points S is not theempty set. We deﬁne that the combined concept (S, V 2 ) of the set of points
S and the 2-dimensional real metric vector space V 2 is the Euclidean plane if
we have the conditions (i), (ii) in the following:(i)(ii)For an arbitrary pair A and B of points in S, we have only one vector a−−→
in V 2 . We express this as a = (AB). Further, for an arbitrary point Ain S and an arbitrary vector a in V 2 , we have only one point B in S so−−→that we have the equality (AB) = a.
−−→−−→
If we have two equalities a = (AB) and b = (BC), we have the equality−→
a + b = (AC).

Then we denote E 2 = (S, V 2 ). We happen to say simply that the Euclidean
plane is the plane.
We say that this is Weyl’s system of axioms.
Definition 1.3 (Point and vector) Assume that E 2 = (S, V 2 ) is theEuclidean plane. We say that a point in S is a point in E 2 and a vector in
V 2 is a vector in E 2 .
Definition 1.4 (Origin and position vector) We ﬁx one point O in
E 2 and we say that this point is the origin．Then, for an arbitrary point P in−−→
E 2 , we say that the vector a = (OP ) is the position vector of the point P
Definition 1.5 (Line segment) Assume that a0 and a are two vectors
in E 2 and we have a ̸= 0. Now we assume that two points A and B are
the points of position vectors a and a0 + a respectively. Then we deﬁne thatthe line segment connecting the points A and B is the set of all points whose
position vectors are the vectors a0 + ta, (0 ≤ t ≤ 1). Then we denote this line
segment as AB. We deﬁne that the length of the line segment AB is AB = ∥a∥.
Here ∥a∥ denotes the norm of a.
−→Definition 1.6 (Straight line) Assume that a = (OA) is a position
2vector of a point A in E and we assume that the vector a in E 2 is not equal
to 0. Then we deﬁne that the set of all points P whose position vectors area0 + ta, (−∞ < t < ∞) is a straight line with the direction vector a.

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Yoshifumi ItoDefinition 1.7 (Triangle) We denote the points whose position vectors
are three vectors a0 , a0 + a1 and a0 + a2 in E 2 as A, B and C respectively.
Then we say that the set of all points whose position vectors are the vectors
a0 + t1 a1 + t2 a2 , (0 ≤ t1 , t2 ≤ 1, 0 ≤ t1 + t2 ≤ 1) is a 2-dimensional triangle
ABC. Further we say that
position vectors are the( the set of all points whose∪∪
vectors a0 +t1 a1 +t2 a2 , (t1 , t2 ) ∈ ([0, 1]×{0} {0}×[0, 1] {(t1 , t2 ); t1 ≥))0, t2 ≥ 0, t1 + t2 = 1}
is a triangle ABC.

Definition 1.8 (Parallel) We say that two straight lines α1 and α2 areparallel if a1 and a2 are linearly dependent when we assume that the direction
vectors of α1 and α2 are a1 and a2 respectively.
Definition 1.9 (Length) For two points A and B in E 2 , the length AB−−→of a directed line segment AB is deﬁned to be the norm ∥a∥ of the vector−−→(AB) = a. We also say that this is the length of the line segment AB.
Definition 1.10 (Angle of two line segments)
line segments is deﬁned to beThe angle α of two

α = cos−1 (a, b)
when we assume that the direction vectors a and b of these line segments are
not equal to the zero vector 0. Here (a, b) denotes the inner product of a andb.
By using Deﬁnition 1.2 ∼ Deﬁnition 1.10, we prove the existence of a modelof Euclidean geometry of the plane in Theorem 1.1.
Theorem 1.1 (Existence of a model of Euclidean geometry) On
the Euclidean plane E 2 = (S, V 2 ), we have the statements (1)∼(5) in thefollowing:(1)

We can draw a line from an arbitrary point to another arbitrary point.

(2)

We can extend continuously the finite line (the line segment) to one
straight line.

(3)

We can draw a circle with an arbitrary point and a distance (the radius).

(4)All right angles are mutually equal.

(5)

If one straight line intersects two straight lines and the sum of two innerangles on the same side is less than the 2-times right angles, then these
two straight lines intersect on the side where the sum of two inner angles
is less than the 2-times right angles when these two straight lines arebelonged infinitely.
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97

This is a model of Euclidean geometry.
Because we proved the existence of a model of Euclidean geometry by theTheorem 1.1, we proved the consistency of the system of axioms of Euclideangeometry.
Thereby we prove that the geometry in Euclid’s principle holds in the Euclidean plane E 2 = (S, V 2 ).

2 Study on the system of axioms of Euclidean
geometry
In this section, we study the system of axioms of Euclidean geometry.
As for the results in this section, we refer to Ito “Fushigi no Izumi(mathematicsI)”, Chapter 14.
What are the mathematical concepts such as the set, the number and Euclidean geometry.
By virtue of the axiomatic method used until now, the system of axioms
is concerned as the ﬁnal base point of the logical thinking in order to provethe derived theorems by the system of axioms. So that, the system of axioms
does not deﬁne the concept of numbers as an ideal existence. The proof of thetheorems of the mathematical concept is the biggest purpose and we consider
the system of axioms as its base point. Then the mathematical concepts, forexamples, the point, the line and the plane in Euclidean geometry are expressedsimply as the undeﬁned terminologies, and the system of axioms only deﬁnes
their relations.
By virtue of my new axiomatic method, the system of axioms is the condition for the deﬁnition of the mathematical concepts and the consistency of thesystem of axioms is proved by virtue of the proof of the existence theorem ofsuch a mathematical concept. All mathematical concepts are the conceptual
existences in the set theory I.
A d-dimensional metric aﬃne space is a certain type of manifold and it is
a ﬂat manifold. A d-dimensional metric vector space is attached at each pointof the manifold.
These d-dimensional metric vector spaces are all isomorphic, and the ddimensional metric vector space at each point is put on each other by theparallel translation.
Then the d-dimensional metric vector space at each point plays the role ofcertain type of the coordinate neighborhood. This is the diﬀerence between ad-dimensional metric vector space as itself and a d-dimensional metric aﬃnespace.

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Yoshifumi Ito

Thereby, the diﬀerential and integral calculus on the d-dimensional metricaﬃne space is realized as the diﬀerential and integral calculus on a manifold.
The diﬀerential and integral calculus is principally realized on a d-dimensional
Euclidean space as a d-dimensional metric vector space.
In the expression of the system of axioms which Euclid gave in his Principle,
the deﬁnitions of the various types of the mathematical concepts are not givenexactly. The point, the line and the plane etc. are considered as the “undeﬁned
terminology” and their relations are deﬁned as the system of axioms. Therefore,
this Principle is considered to be a systematical formulation of the experiencedknowledges of the geometry at that time. If we consider that as a ﬁrst bookof the mathematical results at that time, this consideration is considered as anatural fact.
After this, the deﬁnitions of the concepts of the natural numbers and thereal numbers were the systematical formulation of the experienced knowledges
about the natural numbers and the real numbers. They are not the mathematically exact deﬁnitions. Recently I complete the deﬁnitions of the naturalnumbers and the real numbers and the proofs of their existence theorems byusing my new axiomatic method.
In the deﬁnition of the system of axioms of Euclidean geometry given byEuclid himself, Euclid deﬁnes the point, the straight line and the plane as theundeﬁned terminology and prescribe their relations as the system of axioms.
In Hilbert’s system of axioms of Euclidean geometry, he deﬁned the point, thestraight line and the plane as the undetermined terminology and prescribed
their relations as the system of axioms. At this point, Hilbert’s system of
axioms of Euclidean geometry is essentially the same thing as Euclid’s systemof axioms.
In this time, we consider that the plane is the 2-dimensional metric aﬃnespace in the set theory I. There, the point, the straight line are deﬁned correctlyas the sets of points in the plane. Then we adapt the Euclid’s system of axioms
as the relations satisﬁed by the point, the straight line and the plane. Thereby
the point, the straight line and the plane are deﬁned as the natural existenceof the set theory I and they are deﬁned without any ambiguity. Euclidean
geometry has the natural foundation as the existence in the set theory I.
Euclidean plane, namely the 2-dimensional metric aﬃne space and Euclidean geometry are not identical. Euclidean geometry is the mathematicsof the ﬁgures on the plane. On the Euclidean plane, we can study not only Euclidean geometry but also the diﬀerential and integral calculus and the theoriesof physical phenomena.
The exact deﬁnition of the concepts of the point, the straight line and theplane is not only the problem of Euclidean geometry but also the problemof the exact deﬁnition of Euclidean plane and the exact determination of itsstructure. The problem of Euclidean geometry is the study of the propertiesof the ﬁgures drawn by using the ruler and compass on the Euclidean plane.
When we study the Euclidean geometry on the plane, we consider the planar
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ﬁgures which we can draw by using the ruler and the compass. The experienced
knowledge is that these planar ﬁgures are the ﬁgures composed of the points
and the line segments in the plane. We have not seen the ﬁgures drawn by
using the chair, the desk and the beer jokey as Hilbert’s said.
This is the experienced knowledge. When we study them logically, westudy them by formulating these experienced knowledges systematically byusing the system of axioms. The nonsense idea of Hilbert’s consideration does
not appear. Anywhere, This means the biggest mistakes because he formalize
the mathematics as the system of only the nonsense symbolic calculation.
If we consider the extremely abstract formalization, the considered fact
becomes nonsense.
What form is the “Triangle” drawn by using the chair, the desk and thebeer jokey as Hilbert said.
Since the word does not express any meaning, we cannot understand themeaning of that word. Why we can study the mathematics by using Hilbert’sidea. Even if we can understand the Euclidean geometry by using the meaningless symbols and words as the undetermined terminology said by Euclid andHilbert, it does not mean that we understand the geometry purely theoretically.
In any way, he consider by himself that he understand the properties of theﬁgures by axiomatic method on the basis of the accumulation of the experiencedknowledges.
Thus, the understanding of the expression by using the meaningless symbols
and words is considered as the misunderstanding by virtue of those considerations.

References[1]

D. Hilbert, Hilbert’s Problems of Mathematics, —Hilbert’s Problem —,Kyoritu Syuppann, 1979. (In Japanese). (Translation and Explanation
by S.Hitotumatu).

[2]

Yoshifumi Ito, Introduction to Linear Algebla, Kyoritu Syuppann, 1980.
(In Japanese). (Out of print).

[3]————-, Axioms of Arithmetic, Science House, 1999. (In Japanese).

[4]————-, (Shared writer, Y. Ito), Iwanami Dictionary of Mathematics,
4th ed. Ed. by Japanese Society of Mathematics, Iwanami Shoten, 2007.
(In Japanese).

[5]————–, The Principle of Natural Philosophy, preprint, 2012. (InJapanese).

[6]————–, Set and Topology, preprint, 2013. (In Japanese).
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Yoshifumi Ito[7]————-, Axiomatic Set Theory, preprint, 2020. (In Japanese).

[8]———, Foundation of Linear Algebra, preprint, 2021. (Especially, Section 16.5, Proof of the consistency of the system of axioms of Euclidean
geometry). (In Japanese).

[9]—————, Fushigi no Izumi,(Mathematics I), Chapter 4, preprint. (InJapanese).

[10]

Japanese Society of Mathematics (Ed), Iwanami Dictionary of Mathematics, 3rd. ed., Iwanami Syoten, 1985. (In Japanese).

[11]—————, Iwanami Dictionary of Mathematics, 4th ed., Iwanami Shoten,2007. (In Japanese).

[12]

J. Moriwaki, Foundation of Plane Geometry, Science Co, 2021. (InJapanese).

[13]

Euclid, Euclid’s Principle, Kyoritsu Shuppan. (In Japanese). (Translated by K.Nakamura and others).

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