Originally posted by Traveling Again
Infinity minus 1 is pretty darn close.
heinzkat's post in opposition to this statement is at its heart true, but in my humble opinion uses the same flawed framing of infinity that Traveling Again's statement uses incorrectly. The common misconception to treat "infinity" as a
number is in itself incorrect and leads to the false supposition that the concept "infinity minus one" is a number one away from "infinity."
but infinity really, i would say, is a
limit. or in some cases, a "direction." it is in its essence innumerable and by definition unquantifiable. hence the concept of a number [infinity - 1] doesn't really make any sense.
what heinzkat really means to say i think is that if you take a finite quantity of things away from an infinite number of things, you still have an infinite number of things left. another way of seeing it would be to say that any real finite number (no matter how incredibly large, or perhaps even conceptually daunting) has an infinite number of real numbers larger than it.
but the question of whether 2 is closer to infinity than 1 is an interesting one... and i would argue that, in fact, it is no closer, as have others in this thread before me. consider the fact that the set A={1,2,3,4,...} and the set B={1,4,9,16,...} are exactly the same size even though B is a subset of A! it seems that our intuition about counting things often fails us when dealing with things that are by definition not bounded by finite conception.
imagine two gigantic piles of apples and oranges. if we were to compare their size, we COULD try to count all the apples, THEN count all the oranges, and compare the numbers. but another way would be to grab an apple and an orange, and put them aside. then repeat over and over until one pile is gone and the other has leftovers - the one with leftovers would be larger. if there are none left over, then the two piles were the same size. this is akin to creating a one-to-one relationship between two sets. so then to answer the question of whether 2 is closer to infinity than 1, if we can map the set G = {2,3,4,...} to the set H={3,4,5,...} in a one-to-one relationship, then the sets will be of equal size. in other words, there will be just as many numbers larger than one as there are larger than two, and so 1 and 2 will be equally close to infinity.
but the one-to-one relationship is relatively easy to define. for any number n in set G, associate it with the number (n+1) in H. every number in G has a corresponding "buddy" in H, and so the two sets are of equal size. hence, 2 is no closer to infinity than 1.
whew! long-winded post i hope i didn't scare anyone (or piss anyone) off! 🙂
possible number of moves single chess match?
08 Dec '09 22:37
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