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Boxes, and the Art of Thinking Outside of Them
Posted:
Jul 7, 2008 7:44 AM


Greetings, All.
I'd like to introduce you to Box Theory: a geometricallybased theory of sets. The beauty of this theory is twofold. First, all denoted boxes within a boxset must comport to the physical laws of the phenomenal world, as they're intended to remain applicable to objects outside of the realm of mere mathematics. Second, but no less importantly, box theory sets have containment values that comport to ordinary numbers. This means that a denotation such as '{{},{},{},{},{},{},{},{},{},{}}' can be simplified thus: '{10e}', with "10" denoting the amount of contained boxes, and "e" denoting the fact that all 10 of them are empty. Moreover, the denotation '{{{}},{{}},{{}},{{}},{{}}}' can be written so: '{10: 5e/5o}', meaning the encompassing set has a total containmentvalue of 10: five empty; five occupied. I believe these aspects (especially the latter) render the theory more readily accessible to nonspecialists.
Now, the following examples are intended to highlight an interesting correlation: something I discovered only yesterday. Consider {56: 30e/26o} :if you opened this encompassing box, you'd see only 30 smaller boxes inside, because 26 would be hidden by virtue of subcontainment. What's more, of the 30 immediately visible boxes, only four would be empty, as the other 26 empties would be residents of 26 of the visible, occupied boxes. Now consider the converse approach, in which the amount of contained empties is considerably fewer than their occupied counterparts. Upon opening {43: 2e/41o}, you'd immediately see only two smaller boxes inside. One of these immediately visible boxes would be empty; the other would be occupied with boxes within boxes adding up to the remaining 41, the innermost subcontained box being the other empty. The point here is that the respective amounts of empty boxes in the above boxsets correspond to the numbers of boxes that are "immediately visible" when the encompassing boxes are opened (and, FYI, this holds true in every case I've tried).
I'm open to discussion on the theory in general, but more specifically to suggestions in terms of naming and explaining how and why the above correlation occurs.
Thanks.



