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Axioms:

(1) Hilbert's axioms of incidence

(2) Hilbert's axioms of order

(3) Definition of length for a smooth curve

(4) Given a plane and a line lying in the plane, we can find at least one point lying in the given plane but not in the given line.

(5) For every pair of points $P$ and $Q$ there exists a non-negative real number $PQ$, called the distance from $P$ to $Q$. $PQ=0$ if and only if $P$ and $Q$ refer to the same point.

(6) For each line $l$ there is a one-to-one correspondence from $l$ to $R$ such that if $P$ and $Q$ are points on the line that correspond to the real numbers $x$ and $y$, respectively, then $PQ=|x−y|$.

(7) Every segment can be laid off upon a given side of a given point of a given straight line in exactly one way.

(8) Hilbert's axiom III.2 and axiom III.3

(9) Let $a$ be any line and $A$ a point not on it. Then there is one and only one line in the plane, determined by $a$ and $A$, that passes through $A$ and does not intersect $a$.

(10) Dedekind's axiom

(11) Definition of area

(12) For any three pairwise distinct points $A$, $B$ and $C$, we have $AB+BC \geq AC$ with equality only when $B$ lies between $A$ and $C$.

This axiomatic system differs from Hilbert's axiomatic system mainly in that Hilbert's axioms regarding angles is discarded. The most potential substitution is (12).

(3), (5) and (11) may be unimportant in an axiomatic system.

(4) may be derived from (1) and (2).

(6) and (10) may be derived from Hilbert's axioms of continuity.

(7) and (9) are very similar to Hilbert's statements. However, (7) ensures uniqueness and (9) ensures existence.

Are these axioms qualified to serve as an alternative to Hilbert's axioms? Probably the answer is no. But I can't construct a non-Euclidean model which satisfies all axioms (taxicab geometry is similar). Any reference is cordially appreciated.

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  • $\begingroup$ If you are willing to add the notion of length for smooth functions (defined exactly how?; The standard definitions use the notion of a vector space), then you can as well just use Riemannian metric, etc. $\endgroup$ – Moishe Kohan Jun 27 '17 at 22:47
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    $\begingroup$ You axiom 5 assume the real numbers, so you already have (10). If you have the real numbers you do need more axioms, you can just define the cartesian plane over the real numbers. $\endgroup$ – Julien Narboux Jun 28 '17 at 11:39

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