| Subject: Wade was right (sort of) |
Author:
Duane
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Date Posted: 05/25/04 11:03am
Wade:
That last thread got too cluttered, so I started a new one.
After a little more thought, I realized the following about your first premise (that there is a 1 to 1 correspondence of days to years).
You defined time as a set in which each element corresponds to 365 distinct non-self elements that are (arbitrarily) less than the element in question, within that same set.
Take the empty set - now add one element. There's a day, so he spends 365 more days writing about it. At the end of that year, as we've added individual elements to the set, we end up with 366 individual elements.
Now we've satisfied your initial condition, that day 0 has a one to one correspondence to year 0 (days 1-366). But, in doing so, we've added 365 more days that also must have, each 365 distinct, greater elements to correspond to each, so your condition (1 to 1 correspondence) isn't true.
The 1 to 1 correspondence is not true for any finite subset of natural numbers, and I claim we may state that it is untrue for an infinite set of natural numbers.
This is so because it is not true for the basis case (1 element in the set), nor for the first recursive step (when we add 365 days), nor the second, etc., etc. This satisfies the requirements for proof by induction, and shows that the property (i.e., the 1 to 1 correspondence of days to years) is not true, ever.
I think that your premise appears to be true, intuitively, for the infinite set (infinity / infinity = 1), but that's just illustrating an interesting property of infinity, and ignores the nature of this particular set (as explained above).
Let me know what you think.
Duane
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