Hilbert's axioms. Betweenness and ordered fields.

• Jul 20th 2012, 12:15 PM
abhishekkgp
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 $\displaystyle F$ is a field, and if there is a notion of betweenness in the Cartesian plane $\displaystyle \Pi_F$ satisfying Hilbert's axioms (B1)-(B4), then $\displaystyle F$ must be an ordered field.

The proof in the book reads as follows:

Suppose that $\displaystyle F$is a field and there is a notion of betweenness in the plane $\displaystyle \Pi_F$ satisfying (B1)-(B4). We define the subset $\displaystyle P \subset F$ to consist of all $\displaystyle a \in F$ such that the point $\displaystyle (a,0)$ of the x-axis is on the same side of $\displaystyle 0$ as $\displaystyle 1$.
"Now one can easily show that $\displaystyle a,b \in P \Rightarrow a+b \in P$."

... 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 $\displaystyle (0,0)*(1,0)*(a,0) \Rightarrow (0,0)*(0,1)*(0,a)$ .But now I am stuck. Can someone help.
• Jul 20th 2012, 12:55 PM
Plato
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 $\displaystyle F$ is a field, and if there is a notion of betweenness in the Cartesian plane $\displaystyle \Pi_F$ satisfying Hilbert's axioms (B1)-(B4), then $\displaystyle F$ must be an ordered field.
The proof in the book reads as follows:
Suppose that $\displaystyle F$is a field and there is a notion of betweenness in the plane $\displaystyle \Pi_F$ satisfying (B1)-(B4). We define the subset $\displaystyle P \subset F$ to consist of all $\displaystyle a \in F$ such that the point $\displaystyle (a,0)$ of the x-axis is on the same side of $\displaystyle 0$ as $\displaystyle 1$.
"Now one can easily show that $\displaystyle a,b \in P \Rightarrow a+b \in P$."
... 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 $\displaystyle (0,0)*(1,0)*(a,0) \Rightarrow (0,0)*(0,1)*(0,a)$ .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?
• Jul 20th 2012, 01:11 PM
abhishekkgp
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 $\displaystyle l$ be a line not containing any of A,B,C. If $\displaystyle l$ 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 $\displaystyle a+b$, the '+' here is the field addition and $\displaystyle a,b \in F$.
• Jul 20th 2012, 02:37 PM
Plato
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 $\displaystyle l$ be a line not containing any of A,B,C. If $\displaystyle l$ 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 $\displaystyle a+b$, the '+' here is the field addition and $\displaystyle a,b \in F$.

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 $\displaystyle ,\mathcal{P},$ in an ordered field.

From your OP, I assumed that the set $\displaystyle \mathcal{P}$ 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 $\displaystyle +$ operation because it seems to me that you are required to show that $\displaystyle (a+b.0)$ is in the 1-side of zero.

Sorry, but I simply do not understand you notation.
• Jul 20th 2012, 10:25 PM
abhishekkgp
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 $\displaystyle ,\mathcal{P},$ in an ordered field.

From your OP, I assumed that the set $\displaystyle \mathcal{P}$ 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 $\displaystyle +$ operation because it seems to me that you are required to show that $\displaystyle (a+b.0)$ is in the 1-side of zero.

Sorry, but I simply do not understand you notation.

Does the following make sense?

If $\displaystyle F$ is a field, and if there is a notion of betweenness in the Cartesian plane $\displaystyle \Pi_F$ satisfying Hilbert's axioms (B1)-(B4), then $\displaystyle F$ must be an ordered field.
Here $\displaystyle \Pi_F$ is the cartesian plane on the field $\displaystyle F$, that is, the set of all ordered pairs $\displaystyle (x,y)$, where $\displaystyle x,y \in F$.