Saturday, December 18, 2010

The Infinite Set of Zeros

Proposition: There exist an infinite number of zeros, and some zeros are greater than others.

Imagine you were playing Russian Roulette with a gun that, instead of having 6 bullet slots, had 666,666,666,666,666,666,666,666,666,666,666,666 bullet slots. And one bullet. Would you be scared? The chance of being shot by that one bullet is very low, in fact, so low that it is zero for all practical purposes. Now, suppose the same game were to be played, but with six hundred and sixty six bullets instead of one. Your chance of dying goes up six hundred and sixty six times, though it is still practically zero. Now, imagine that the shooter opens up the gun in front of you, and removes EVERY bullet. Then, he proceeds to play Russian Roulette with you. Would you be scared any more?

I do not know about you, but I would be most scared in the second case, and absolutely not scared in the last case. The reason is obvious enough: we do know that the odds, though practically zero in all cases, are different in each case.

Georg Cantor was a mathematician who proved that there exist an infinite number of infinities, and some infinities are greater than others. The crux of his work is explained very lucidly by Steven Strogatz here:

Cantor [proved] that there are exactly as many positive fractions (ratios p/q of positive whole numbers p and q) as there are natural numbers (1, 2, 3, 4, …).  That’s a much stronger statement than saying both sets are infinite.  It says they are infinite to precisely the same extent, in the sense that a “one-to-one correspondence” can be established between them.

If many infinities exist and are unequal, their reciprocals, which are zero must also be unequal. That is, if X and Y are two positive infinities such that X<Y, then 1/X>1/Y. And there are an infinite number of such infinities. Thus, it must follow that there exist an infinite number of zeros, which are the reciprocals of these infinities. Each of these is different from the absolute zero, denoted by 0.

Note to readers: There is probably a fallacy in this argument. Do let me know your views in comments...