Appendix 2
Volumes
This appendix involves volumes of sets in E3. Its purpose is to show
that the theory of volumes is not simple. We shall reveal the fact that
there is something inherently contradictory in the combination of the
following four assumptions:
(A,) Each bounded set S in E3 has a volume.
(A2) If Sl and S2 are congruent bounded sets in E3, then Sl and S2
have equal volumes.
(A3) If a bounded set S in E3 is composed of two or three or four or five
separate and distinct subsets, then the volume of S is the sum of
the volumes of its subsets.
(A4) If S is a solid spherical shell having inner and outer radii for which
0 < rl < r2, then the volume of S is positive.
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