Please refer the following book for this question: David Hilbert, The Foundations of Geometry, English Translation by E. J. Townsend, Reprint Edition, The Open Court Publishing Company, 1950.

In this book the axiom II. 1 (Page 4) states:

II, 1. If A, B, C are points of a straight line and B lies between A and C, then B lies also between C and A.

My question: In the above statement, what does the words "lies between" mean ? If we take the obvious meaning, then is it not clear that the statements"B lies between A and C" and "B lies between C and A" are same ? Then why should we treat this as an axiom ?

Thanks !

  • $\begingroup$ It needs to be an axiom because, while it is "clear" that these two statements are the same, there is no way to prove that these two statements are equivalent (without formally defining what it means for $B$ to "lie between" two points, which I doubt they want to do). $\endgroup$ – Morgan Rodgers Aug 13 '16 at 13:40
  • $\begingroup$ @MorganRodgers, So, the fundamental question remains: what do we mean by "lie between" ? This term is not made clear in the book. Then how does this statement make sense? $\endgroup$ – Rajkumar Aug 13 '16 at 13:47
  • $\begingroup$ It's meant to be an intuitive concept. We naively understand what it means for a point to be between two others, so we avoid defining it explicitly (which would be complicated and somewhat pointless). Instead we are given that we have a concept of "betweenness" for points on a line, and some basic axioms for how this relationship behaves. $\endgroup$ – Morgan Rodgers Aug 13 '16 at 13:52
  • $\begingroup$ Still this is not clear to me. The intuitive concept of "betweenness" is what stated in this axiom. So, why this axiom ? $\endgroup$ – Rajkumar Aug 13 '16 at 13:57

In Hilbert's axioms, "lies between" is a primitive notion. This means that it is not given any definition or meaning besides the fact that it is some ternary relation between points, and the only things you can say about it are the things that are stated as axioms. So you cannot make any assumptions based on its "intuitive" meaning; everything about this notion must either be stated explicitly as an axiom or follow logically from the axioms. If you find this confusing, you might find it helpful to change the name to something meaningless: instead of saying "$B$ lies between $A$ and $C$", say "the triple $(B,A,C)$ is glumpy". There's no reason to think that if $(B,A,C)$ is glumpy then $(B,C,A)$ has to also be glumpy, so if we want to be able to assert that it is true we need to assume it as an axiom. The only reason it is traditionally called "lies between" instead of "glumpy" is that it is supposed to model our intuitive geometric idea of "lying between".

  • $\begingroup$ Perhaps, the answer of following question will make the concept more clear: Is there any concrete example of geometry based on all the Hilbert's axiom except the above axiom in which the statement "B lies between A and C" does not imply "B lies between C and A" ? $\endgroup$ – Rajkumar Aug 13 '16 at 15:28
  • $\begingroup$ +1 partly because good answer, and partly because "glumpy". $\endgroup$ – Noah Schweber Aug 13 '16 at 15:46
  • $\begingroup$ @Rajkumar: Probably, but I don't know one off the top of my head (it is complicated to find one since there are so many other axioms). $\endgroup$ – Eric Wofsey Aug 13 '16 at 15:59

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