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→Physically impossible?
:The difficulty, of course, comes with the difference between "mathematical objects" and physical ones. I would say it's not just physically impossible with real objects, but ''logically'' impossible as well. The cuts required for a paradoxical decomposition are literally infinitely small and precise. Since real objects are made of indivisible particles which cannot be divided (quarks, electrons), and because their positions cannot be known with any great accuracy at that scale, the cuts could never be performed - it's not only a physical impossibility, but also a logical impossibility for any object made of a finite number of particles of non-zero size.
:The difficulty, of course, comes with the difference between "mathematical objects" and physical ones. I would say it's not just physically impossible with real objects, but ''logically'' impossible as well. The cuts required for a paradoxical decomposition are literally infinitely small and precise. Since real objects are made of indivisible particles which cannot be divided (quarks, electrons), and because their positions cannot be known with any great accuracy at that scale, the cuts could never be performed - it's not only a physical impossibility, but also a logical impossibility for any object made of a finite number of particles of non-zero size.
:With a "mathematical object," the case is of course different - these are more mental constructs than anything else, which we picture as being composed of infinitely many points, which are infinitely small. In order to perform a cut, we need only describe that cut. So in this situation, it IS logically possible. [[User:JacobB|JacobB]]<sub>[[User talk:JacobB|Shout out!]]</sub> 23:09, 6 June 2010 (EDT)
:With a "mathematical object," the case is of course different - these are more mental constructs than anything else, which we picture as being composed of infinitely many points, which are infinitely small. In order to perform a cut, we need only describe that cut. So in this situation, it IS logically possible. [[User:JacobB|JacobB]]<sub>[[User talk:JacobB|Shout out!]]</sub> 23:09, 6 June 2010 (EDT)
::If you wanted to discuss this further, I'd be more than happy to here, or by phone or email. [[User:JacobB|JacobB]]<sub>[[User talk:JacobB|Shout out!]]</sub> 23:10, 6 June 2010 (EDT)