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Hilbert's Betweenness Axioms

    A purpose of the Hilbert Betweenness Axioms (Hilbert's Order Axioms) is to give meaning to the undefined term between; seeing as between is undefined, we have only the axioms to guarantee that the points in this geometry behave in a way that is consistent with our interpretation of "betweenness". This topic points out that there are exactly two sides to a line in the Euclidean plane (Half-Planes Proposition), and that lines are not circular (Betweenness Property), and how to decompose a line into its parts (Linear Decomposition and Line Separation Propositions). The interior of an angle and the Crossbar Proposition are also detailed. Some of these results were taken for granted by Euclid.

The notation will be used to denote that is between and

Axiom (Betweenness Axioms) The following axioms are called the Betweenness Axioms.

    (i) (Linearity) If , then and are three collinear points and
    
    (ii) (Extension) Given any two distinct points and , there exist a point lying on such that
    
    (iii) (Order) If and are three distinct points on the same line, then one and only one of the points is between the other two.
    
    (iv) (Separation) Given any line and any three points and not lying on . If and are on the same side of and and are on the same side of then and are on the same side of If and are on opposite sides of and and are on opposite sides of   then and are on the same side of

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