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Math Help - Hilbert's axioms. Betweenness and ordered fields.

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    Senior Member abhishekkgp's Avatar
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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 F is a field, and if there is a notion of betweenness in the Cartesian plane \Pi_F satisfying Hilbert's axioms (B1)-(B4), then F must be an ordered field.



    The proof in the book reads as follows:

    Suppose that Fis a field and there is a notion of betweenness in the plane \Pi_F satisfying (B1)-(B4). We define the subset P \subset F to consist of all a \in F such that the point  (a,0) of the x-axis is on the same side of 0 as 1.
    "Now one can easily show that 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  (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 F is a field, and if there is a notion of betweenness in the Cartesian plane \Pi_F satisfying Hilbert's axioms (B1)-(B4), then F must be an ordered field.
    The proof in the book reads as follows:
    Suppose that Fis a field and there is a notion of betweenness in the plane \Pi_F satisfying (B1)-(B4). We define the subset P \subset F to consist of all a \in F such that the point  (a,0) of the x-axis is on the same side of 0 as 1.
    "Now one can easily show that 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  (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 l be a line not containing any of A,B,C. If 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 a+b, the '+' here is the field addition and 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 l be a line not containing any of A,B,C. If 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 a+b, the '+' here is the field addition and 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 ,\mathcal{P}, in an ordered field.

    From your OP, I assumed that the set \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 + operation because it seems to me that you are required to show that (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 ,\mathcal{P}, in an ordered field.

    From your OP, I assumed that the set \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 + operation because it seems to me that you are required to show that (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 F is a field, and if there is a notion of betweenness in the Cartesian plane \Pi_F satisfying Hilbert's axioms (B1)-(B4), then F must be an ordered field.
    Here \Pi_F is the cartesian plane on the field F, that is, the set of all ordered pairs (x,y), where x,y \in F.
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