I am reading the book "Geometry: Euclid and Beyond - Robin Hartshorne". Here's the first half of Proposition 15.3 from the book.
If is a field, and if there is a notion of betweenness in the Cartesian plane satisfying Hilbert's axioms (B1)-(B4), then must be an ordered field.
The proof in the book reads as follows:
Suppose that is a field and there is a notion of betweenness in the plane satisfying (B1)-(B4). We define the subset to consist of all such that the point of the x-axis is on the same side of as .
"Now one can easily show that ."
... Which I am not able to show and so I need your help.
I was able to prove, using Pasch's axiom of betweenness, a.k.a (B4) in the book, that .But now I am stuck. Can someone help.
B1) If B is between A and C, (written A*B*C), then A, B, C are three distinct points on a line, and also C*B*A.
B2) For any two distinct points A, B, there exists a point C such that A*B*C
B3) Given three distinct points on a line, one and only one of them is between the other two.
B4) Let A, B, C be three non-collinear points, and let be a line not containing any of A,B,C. If contains a point D lying between A and B, then it must also contain either point lying between A and C or a point lying between B and C, but not both.
As for , the '+' here is the field addition and .
Well you have succeeded in confusing me. I taught the axiomatic geometry many times and hence Hilbert's betweenness axioms. I have never seen them applied to the set of positives in an ordered field.
From your OP, I assumed that the set was defined a half line in the x-axis. In which case, it is a convex set and the result should follow. I asked about operation because it seems to me that you are required to show that is in the 1-side of zero.
Sorry, but I simply do not understand you notation.