Hilbert's axioms. Betweenness and ordered fields.

In summary, the conversation is about the book "Geometry: Euclid and Beyond - Robin Hartshorne" and Proposition 15.3. The proof in the book shows that if a field $F$ has a notion of betweenness in the Cartesian plane $\Pi_F$ satisfying Hilbert's axioms, then $F$ must be an ordered field. The speaker is stuck on one part of the proof and is asking for help. Another person suggests using Pasch's axiom to show that $a,b \in P \Rightarrow a+b \in P$, using the points $(0,0), (1,0), (a,0)$ and $(b,0)$. With this, the speaker is able to complete the
  • #1
caffeinemachine
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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 $F$ is 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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  • #2
To show that $a,b \in P \Rightarrow a+b \in P$, consider the points $(0,0), (1,0), (a,0)$ and $(b,0)$. By assumption, these four points are all on the same side of the y-axis. By Pasch's axiom, it follows that $(0,0)*(1,0)*(a+b,0)$ is also on the same side of the y-axis. Since $(1,0)$ and $(a+b,0)$ are on the same side of $0$, we have that $a+b \in P$, so $a, b \in P \Rightarrow a+b \in P$ as desired.
 

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

What are Hilbert's axioms?

Hilbert's axioms are a set of 20 mathematical principles that serve as the foundation for Euclidean geometry. These axioms establish the basic properties of points, lines, and planes, and are used to prove theorems and construct geometric shapes.

What is "betweenness" in the context of Hilbert's axioms?

In Hilbert's axioms, betweenness refers to the concept of a point lying between two other points on a line. This is one of the fundamental properties of a line according to Hilbert's axioms.

How are Hilbert's axioms related to ordered fields?

Hilbert's axioms are related to ordered fields in that they both involve the concept of order and relations between objects. In ordered fields, numbers are arranged in a specific order (e.g. from smallest to largest), while in Hilbert's axioms, points and lines are also arranged in a specific order (e.g. betweenness).

What are some applications of Hilbert's axioms?

Hilbert's axioms have been used in various fields such as geometry, computer graphics, and game theory. They provide a rigorous framework for understanding and analyzing geometric concepts and are essential for proving theorems and solving problems in these fields.

Are Hilbert's axioms still relevant today?

Yes, Hilbert's axioms are still relevant today and are considered to be one of the foundations of modern geometry. They are used in various fields of mathematics and have been expanded upon and adapted for different purposes, making them a fundamental part of mathematical thinking and problem-solving.

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