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"If all points of a straight line fall into two classes, such that every point of the first class lies to the left of any point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions"

Two questions -

1. Is this still a fundamental axiom in some or all forms of mathematics?

2. How is the inherent self-contradiction resolved?