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

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

    Quote Originally Posted by abhishekkgp View Post
    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?
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    Re: Hilbert's axioms. Betweenness and ordered fields.

    Quote Originally Posted by Plato View Post
    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 $\displaystyle a,b \in F$.
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    Re: Hilbert's axioms. Betweenness and ordered fields.

    Quote Originally Posted by abhishekkgp View Post
    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.
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    Re: Hilbert's axioms. Betweenness and ordered fields.

    Quote Originally Posted by Plato View Post
    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 $\displaystyle x,y \in F$.
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