Hilbert's axioms. Betweenness and ordered fields.

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.

Re: Hilbert's axioms. Betweenness and ordered fields.

Quote:

Originally Posted by

**abhishekkgp** 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.

I think you are going have to list Hartshorne's version of Hilbert' (B1)-(B4),

Also how is a+b defined?

Re: Hilbert's axioms. Betweenness and ordered fields.

Quote:

Originally Posted by

**Plato** I think you are going have to list Hartshorne's version of Hilbert' (B1)-(B4),

Also how is a+b defined?

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 .

Re: Hilbert's axioms. Betweenness and ordered fields.

Quote:

Originally Posted by

**abhishekkgp** 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.

Re: Hilbert's axioms. Betweenness and ordered fields.

Quote:

Originally Posted by

**Plato** 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.

Does the following make sense?

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.

Here is the cartesian plane on the field , that is, the set of all ordered pairs , where .