Prove that the SMSG axiomatic set is not independent.
SMSG Axioms:
Postulate 1. Given any two distinct pointsthere is exactly one line that contains them.
Postulate 2. Distance Postulate. To every pair ofdistinct points there corresponds a unique positive number. Thisnumber is called the distance between the two points.
Postulate 3. Ruler Postulate. The points of a linecan be placed in a correspondence with the real numbers suchthat:
To every point of the line there corresponds exactly one realnumber.
To every real number there corresponds exactly one point of theline.
The distance between two distinct points is the absolute valueof the difference of the corresponding real numbers.
Postulate 4. Ruler Placement Postulate Giventwo points P and Q of a line, the coordinate system can be chosenin such a way that the coordinate of P is zero and the coordinateof Q is positive.
Postulate 5.
Every plane contains at least three non-collinear points.
Space contains at least four non-coplanar points.
Postulate 6. If two points lie in a plane, thenthe line containing these points lies in the same plane.
Postulate 7. Any three points lie in at least oneplane, and any three non-collinear points lie in exactly oneplane.
Postulate 8. If two planes intersect, then thatintersection is a line.
Postulate 9. Plane Separation Postulate. Given aline and a plane containing it, the points of the plane that do notlie on the line form two sets such that:
each of the sets is convex
if P is in one set and Q is in the other, then segment PQintersects the line.
Postulate 10. Space Separation Postulate. Thepoints of space that do not lie in a given plane form two sets suchthat:
Each of the sets is convex.
If P is in one set and Q is in the other, then segment PQintersects the plane.
Postulate 11. Angle Measurement Postulate. Toevery angle there corresponds a real number between 0° and180°.
Postulate 12. Angle Construction Postulate. Let ABbe a ray on the edge of the half-plane H. For every r between 0 and180 there is exactly one ray AP, with P in H such thatm?PAB=r.
Postulate 13. Angle Addition Postulate. If D is apoint in the interior of ?BAC, then m?BAC = m?BAD + m?DAC.
Postulate 14. Supplement Postulate. If two anglesform a linear pair, then they are supplementary.
Postulate 15. SAS Postulate. Given a one-to-onecorrespondence between two triangles (or between a triangle anditself). If two sides nd the included angle of the first triangleare congruent to the corresponding parts of the second triangle,then the correspondence is a congruence.
Postulate 16. Parallel Postulate. Through a givenexternal point there is at most one line parallel to a givenline.
Postulate 17. To every polygonal region therecorresponds a unique positive real number called its area.
Postulate 18. If two triangles are congruent, thenthe triangular regions have the same area.
Postulate 19. Suppose that the region R is theunion of two regions R1 and R2. If R1 and R2 intersect at most in afinite number of segments and points, then the area of R is the sumof the areas of R1 and R2.
Postulate 20. The area of a rectangle is theproduct of the length of its and the length of its altitude.
Postulate 21. The volume of a rectangleparallelpiped is equal to the product of the length of its altitudeand the area of its base.
Postulate 22. Cavalieri's Principle. Given twosolids and a plane. If for every plane that intersects the solidsand is parallel to the given plane the two intersections determineregions that have the same area, then the two solids have the samevolume.
