Why is the line/plane separation postulate necessary?

Asked Sep 27 '23 at 14:34

Active Sep 27 '23 at 14:34

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Usually when we introduce postulates it is so they are of some use to us. I do not see the reason for these two.

The Line Separation Postulate: Each point on a given line divides the line into three disjoint sets: the set containing the point, and the two half lines on either side of the point.

The Plane Separation Postulate: Each line within any plane divides the plane into three disjoint sets: The line and the two half-planes on either side of the line.

I understand the postulates and what they are saying, but I do not see how they can be used. And I have not yet found a proof that uses them in the book.

I guess I am not convinced of their fruitfulness so if anyone can help convince me these are useful/necessary postulates that would be of great help.

asked Sep 27 '23 at 14:34

Chris Christopherson

1,018

Which book? I seem to recall identifying that one in a book on non-Euclidean geometry. – TurlocTheRed Sep 27 '23 at 14:38
Geometry, Bob Jones University Press. It's a high school book. – Chris Christopherson Sep 27 '23 at 14:40
See the Missing Strip Plane and Pasch's axiom. – Mauro ALLEGRANZA Sep 27 '23 at 14:43
See also Are there simple models of Euclid's postulates that violate Pasch's theorem or Pasch's axiom? – Mauro ALLEGRANZA Sep 27 '23 at 14:47
Consider the simple case of the line: if the axiom does not hold, a circle may be used to model the line. But a single point on a circle does not divide it in two "separate" parts. – Mauro ALLEGRANZA Sep 27 '23 at 14:51

Why is the line/plane separation postulate necessary?

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