Cantor, was never wrong

An attempt to refute Cantor's diagnolization proof that real numbers are uncountable, would be in vain. Why? People do not believe in his proof philosophically for the reason that it is so simple. It is the inherent fear with bad experiences that we had with lot many proofs that we keep constantly asking for a proof that is either lengthy or complicated to have a rock solid stability, which is again kind of contradictory. People do try to refute Cantor's arguments by showing up a counter example, and this only shows that they would've gone wrong somewhere in understanding the conventions followed for the most abstract form, the infinity.

Sample Function
Recently, myself and my roomie Vinay were investigating the properties of the function presented below:

The trivial constraints on the above function being that,

If one just experiments with this function all we can see is that given any n, it would map _uniquely_ to a real number in the open interval (0, 1). So, cannot this be a mapping from N->(0, 1) that would effectively refute Cantor's proof?

Integers
Behold, there is a hidden flaw in evaluating the above argument. The flaw is that the set that maps to open interval (0, 1) is _not_ an integer. It might sound crazy, or even freaky to a non-analyst but it makes all the more sense if we could see the dynamics of integers.

Integers, as an infinite set, grows down in one direction. So, at any given instance, however long it might be, the number of digits in each of them would be finite. Given a very long integer, we could always say, what the next integer would be, isn't? Now, take for example a number 111111... Is this an integer? No! Because, one could say of many reasons. One of them being that the growth of this number till infinity is on the horizontal direction and when we group all numbers of this form, that would naturally grow vertically too. This gives an explanation why it could be in a one-one map with (0, 1). The function does nothing but remove the period in elements from (0, 1) and to make things unique, the 0s that are prefixed after period are suffixed in case of the mapping set. Hence, 0.0001 would be mapped from 1000. Another reason being that the next number is not known in this case! which fundamentally fails to be in something called as an 'enumerable' group.

Infinity, flavors fly-overs
So, what is this infinity all about? Why is this so mysterious? No one knows precisely, may be that if one knows, we might not be in a finite existential state, probably :) But one amazing thing about infinity is that, there are different flavors of it existing. One is the infinity of the integers, infinity of the reals, infinity of power-set of reals, and so on and on and on. Does the set of all infinities of the sets namely, aleph0, aleph1, etc.. countable? (Might not be trivial to prove). All uncountable sets are of same size since we have no way to count or to size them up. On this regard, one could slightly agree upon the beliefs Jains had over the infinities. According to them there exists four kinds of infinities, to that which is one dimensional as a line, two dimensional as an area, three as a volume, and the fourth as the infinitely infinite cosmos or the eternity.

All those philosophical discussions over the infinite can be read from: [http://en.wikipedia.org/wiki/Infinity_(philosophy)]

Recently i found a paper submitted with similar spirits of refuting the proof: [http://www.arxiv.org/pdf/math/0103124v1]