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 !
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".