SQUARE45

Let

\mathcal{M} = (S, \text{Inc}, \text{Bet}, \text{Dist})

be a mathematical structure, where

S

is the set of points,

\text{Inc} \subseteq S \times S

is the incidence relation,

\text{Bet}

is the betweenness relation, and

\text{Dist}: S \times S \to \mathbb{R}_{\ge 0}

is a metric function. The system of Euclidean Axioms

\mathcal{A}_{E}

is the set of first-order logical statements that define the properties of

\mathcal{M}

. These axioms include:\\ \text{Ax}_1

(Incidence):

\forall A, B \in S, (A \text{ Inc } B) \iff (A = B)

or

(A \text{ and } B \text{ are distinct points})

.\\\text{Ax}_2

(Betweenness):

\forall A, B, C \in S

, if

B

lies between

A

and

C

, then

\text{Dist}(A, C) = \text{Dist}(A, B) + \text{Dist}(B, C)

.\\\text{Ax}_3

(Congruence): The existence of an isometry group

\text{Isom}(S)

such that for any two segments

\overline{AB}

and

\overline{CD}

,

\text{Dist}(A, B) = \text{Dist}(C, D)

implies a rigid transformation mapping one to the other.\\\text{Ax}_4

(Euclidean Parallel Postulate): For any line

L

and any point

P \notin L

, there exists a unique line

L' \subset S

such that

L'

passes through

P

and

L'

is parallel to

L

(i.e.,

L' \cap L = \emptyset

and

L'

maintains a constant distance from

L

).\\\text{Ax}_5

(Completeness/Dimension): The space

is a complete, connected, and simply connected manifold of dimension 2, satisfying the metric properties of

\mathbb{R}^2$.

Axioms