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...

6 comments:

karatalaamalaka said...

The answer lies in the rigorous definitions. When Strogatz says there are two orders of infinity, he refers to it in the context of properties of sets. What is special about the "number" of rational numbers, or the number of "irrational numbers"? The answer is the "number" refers to a property of the sets- the "sets of rational numbers" and "sets of irrational numbers". Can we count real numbers? No we can't. What is the reason we cannot count them, because "counting" is the "process of mapping one-to-one with the set of natural numbers."

Now, consider the claims:

"there are infinitely many natural numbers" = infinity

"there are infinitely many real numbers" = infinity

"and number of real numbers is more than the number of natural numbers"

can we say infinity > infinity?

NO.

because, "count" is a property of the set. and the set of real numbers is "uncountable". i.e., the set of real numbers does not admit the property of "counting".

to rigorously accommodate such apparent contradictions. the theory of "counting" is generalized to the theory of "measures". measures are rigorously defined as functions on a set, which in the literal sense of the term "measure" the "size" of the set. and each measure is defined on an algebra of sets. and measures can be associated with appropriate algebras of sets.

in this sense, "counting" is a measure. and it is a measure defined on the algebras of subsets of natural numbers (or rational numbers).

but, "counting" is not adequate to quantify the "size" of all sets in the universe. an example is, it won't give us much information about sets containing uncountably large "quantities" of real numbers. for example, consider the set of closed interval, S = [0,1]. there are uncountably large "quantities" of real numbers in S. but there are TWO natural numbers, 0 and 1. so how do we account for this? one way to do so is to define the Lebesgue measure. one of the properties of the Lebesgue measure is that for a set which is an interval, such as S = [a,b], the measure, let's call it mu is, mu = b-a. then, the measure of the set [0,1] would be 1, on the algebra of subsets of real numbers. but if i look at the subset {0,1}, then the measure is "0", or for that matter, if one considers {0,1,...,32} (natural numbers) \subset of [0,32] (real), then the Lebesgue measure of {0,1,...,32} is zero, and the Lebesgue measure of [0,32] is 32, indicating that the set of real numbers between 0 and 32 is much larger than the number of natural numbers. that too, in the sense that the size of the set of natural numbers between 0 and 32 is ZERO.

karatalaamalaka said...

the bigger headaches than the apparent contradiction of your problem is the contradictions that come up in integration, and defining probabilities. you can look at, for example, conditional probabilities and the Borel-Kolmogorov paradox.

http://en.wikipedia.org/wiki/Borel%E2%80%93Kolmogorov_paradox

Prithwiraj said...

@karatalaamalaka

let us take an example

if X>0 then 2X>X
lim(X->infty) 1/X =lim(X->infty) 1/2X=0
but lim(X->0)(2X/X) = 2

how do you account for this paradox unless the two limits give us different zeros?

Prithwiraj said...

@karatalaamalaka purely mathematically my proposition is is not tenable, but in actual physical measurements its very relevant. if you have to measure a quantity smaller than the least count of your gauge, irrespective of its true length it is a zero.

e.g. if you have a meter-long stick, 5cm is also zero, and 20 cm is also zero.

Nanga Fakir said...

First of all, there are two kinds of zeros and infinities - of a totally incomparable kind.

One is a 0 and an infinity as elements of the set.

The other is an equivalence class.

The cardinality infinity and the reciprocal of the cardinal infinity are not numbers/points/elements of a set. They are properties OF the set and are actually equivalence classes.

Hence comparison between the two is not justified since they are two different quantities all together.

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