(4) We obtain a geometric model (P, L) from the usual real Euclidean plane R2 as
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(4) We obtain a geometric model (P, L) from the usual real Euclidean plane R2 as follows. Set P R2, and let L denote the usual collection of lines in the real Euclidean plane R2. Define "betweeness" and "congruence" of line segments for (P,L) as fol lows. For each line l E L choose a continous parametrization f R i from the real numbers for this line: l is straight line in the Euclidean plane R2 so there are infinitely many ways to parametize it continuously (think calculus III). Then if r s t in R and A fl (r), B fi(s), C ,fi(t), we define B to be "between" A and C in l, i.e. A B*C. We define the "length"Explanation / Answer
The first thing you can prove about the Cartesian plane is that it satisfies all of Euclid's postulates. So if you have a Euclidean geometry theorem, it is true in the Cartesian plane.
There might, however, be theorems about the Cartesian plane that are not provable in Euclidean geometry.
In mathematics, we aren't actually concerned with the plane as a "real" object that exists in the world, but rather, axiomatic systems that describe objects.
As one commenter above notes, Euclid's axioms are actually deficient for certain purposes. Hilbert came up with a far more thorough axiom system which is much more closely related to the Cartesian geometry.
For any two lines, let lm mean
L=M or L M . The relation has the three defining properties of an equivalence relation. Indeed this relation is
reflexive: since L = Lis a logical axiom.
symmetric: since l =M implies m=L , and L M implies ML
transitive:assume that ML and Lk. If two of the three lines are equal, we
substitute equals to get Mk
. We now assume that all three lines are different.
The two lines M and k are either parallel or they intersect at one point, by Hilbert’s first Proposition. If M and k would intersect at point P, the line l would have two different parallels through P, which is impossible in an affine plane. Therefore M and k are parallel, as to be shown.
A relation that is reflexive, symmetric and transitive is called an equivalence relation.
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