In continuation to the discussion of Cantor's proof on the stone about the cardinalities of sets, we would continue the discussion further. What makes the proof so counter-intuitive? It is hard to believe in the first shot that the set of all natural numbers are equinumerous to the set of all positive even integers. They just seem to disobey the basic intuitive laws of packing. We've take an object A and we break it uniformly into three different pieces. We would never claim that each piece is as big as that of the original object. This is just the same understanding that we learn by experiencing with 'finite' real world objects. The intuition is just a notion that is so subtle, thin and involuntary generated and stored through the experiences and an apriori knowledge source through the genes. So, our system learns that when a whole is broken into parts and if nothing is lost in this process the parts sums up to the whole. If the breaking operation is of 'absorption' type, the sum of parts would be greater than the whole and if it is of 'emission' type, the sum of parts would be lesser than the whole and certainly in other case it must be the same.
So, looking at the set of all natural numbers in isolation, we neither see it as an object nor attach the thing of 'bigness' or 'infiniteness' to it. By introducing yet another set, say, set of all positive even numbers, instead of comparing if they would be the same in cardinalities, we would tend to match if one set holds the other. It is a basic operation that we tend to do. A similarity comparison for all patterns that we see to associate them to something that we've already perceived before.
Counting is much more than that
Yet, all we do is to identify known patterns in both the sets. Moreover, the learning that natural numbers contains all positive even integers is the place where we remind ourself that the natural numbers are a kind of aggregate object which is 'made up of' yet another aggregate object called by even integers. This information is not misleading but the way our system is designed to deal with objects in an abstract notion. It is just sufficient for the abstracting system in us to perceive and associate the set of all natural numbers and the positive even integers. The system has to abstract and only then it could do any kind of transformation over it. The first association that gets created is the feel of stacked object, a kind of wrapping, one wraps another or one gets 'plugged into' another or in some other form that is really a subjective experience to state. This quick abstraction is the reason for the feeling of counter-intuitiveness when we realize the correctness of the proof. We feel so sheepish that we stare at ourself for being dragged by an illusion!
The operation of counting is just much more than finding if something is similar. For finite elements, it would perfectly hold to see if both patterns hold a similar/same association. In case of infinite elements, it cannot be approximated to any kind of finite object. It would just fail! So, our learning experience with real world objects do come to a question of justification in case of an infinitary scale. So, why do we tend to compare the similarity in the sets? Let us look more closer into what we exactly do when a set of natural numbers and a set of all positive even integers are written down.
More than just a number
Here we go, the set S of all natural numbers and the set S' of all positive even integers.
S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, .... }We see that the comparison we tend to do is to see '2' as a 'number' and '4' as another 'number' and so on. There is no notion as 'number' in our mind but some object, a symbol or a 'mood' driving force of a kind or however we experience the subject to be. So, we tend to see what other objects that exists in the first set too. What we tend to do is that we see that the numbers that occur in the S' occur in the S too, moreover, from the learning we have with the numbers, we know the pattern in which the numbers behave. Needless to write the complete set, we know how it would keep increasing and behave in long run.
S' = { 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, .... }
Perception of infinity
The final spice to mix up in this puzzle is the perception that we have about the infinity. For the first time if one had to see an open set [1, inf), of natural numbers, seeing the first few numbers the rest is assumed. We quickly learn with how the first few numbers behave, the rest too 'behave' in the similar way. So, what more can we say? An infinite set is generally perceived as 'running till the last', something that goes on and on like a long rope or a horizontal line, a 'zzhoop' impulsive feeling and because of this undefined 'endness' to this, we simply tend to 'delay' any kind of consequence or behavioral operation over this pattern of infinite. Not to be mis leaded, the same happens with the sets too. We tend to 'delay' the actual comparison that we are required to logically perform and do a nearest available sub-optimal comparison that is applicable only for the finite sets, that is the 'similarity/same' comparison operation. For the pattern that we perceive with set S and S', the sample set that we take is finite to learn the pattern. The operation that we perform is over the finite set that we learnt and we extend the result on the operation in the 'zzhoop' line till the last and hence delay the final result too. So, the result that we apply for the finite sets turns out to be that for a small bunch of patterns from S we could match same patterns in S' but there are some interweaving patterns in S that could not be matched in S' and hence S contains some elements apart from all elements in S'. This when generalized gives the same result since nothing is 'declared' concretely about the infinite collection, the results observed for the smaller bunch is retained. This results in often errored intuitive prerogative.Why is counting other way wrong?
What is exactly wrong in considering S' being a subset of S? There is nothing wrong. Indeed it is true that S' is a subset of S. The operation of counting as we discussed before is much more than just comparison of patterns. Here we go. Let us again observe where we go wrong by matching up 2 with 2, 4 with 4 and so on.
As a tradition, let us imagine a big (infinitely big) pile of objects of type S and another similar infinitely big pile of objects of type S'. How do we count now? Take an object from S', search for the similar object in S, pick it up and throw both of them down. Repeat for all objects in S'. What we've done here is to not pick, objects that have 'oddness'. This means that for every object that we select, we 'do not' select one object. By selecting only the even number we do not select an odd number. The 'selecting' and 'not selecting' both belongs to the realm of counting and that each object for example 2 in S' is grouped with 2 as well as 1 in S. Only that 2 in S goes along and 1 stays apart. Effectively, for every object we paired two object from another set which is a wrong way to count right!
The right way would be to pair one to one and pick two objects and pair them and this is provided by the elementary function f(x) = 2*x.
The whole is never greater than its parts
Let us come back to the actual topic of discussion something about the sum function that we would like to define over the set S and S'. Let [S] be the sum of all objects from S (valued as numbers) and [S'] be the sum of all objects from S'. So, which would be larger? Here again we hit a dilemma. More so to say, the sum is trivially divergent and hence the literal comparison cannot be done straight. Here are two different result that we could hit upon.
1. The sum [S] is greater than [S']. This could be immediately arrived by using the fact that S' belongs to S and every 'other' element of S is positive and hence it could only be greater than [S].
2. Behold, consider the above pairing function that we've used for counting. In that consider that we have paired first n numbers from S and S'. Trivially, the sum [S] is smaller than [S']. In fact it is two times smaller. Moreover, as we keep adding new pairs to our sums, we could find that the [S'] seems to keep the same pace with [S] and stays twice bigger. By our 'zzhoop' notion of extending till infinite, the [S'] would hold to be twice bigger as [S]. Yet another counter-intuitive law that the whole is less than its part, yet no loss has been incurred. More surprisingly, if we take just the squares alone, they would yield a much larger sum than the [S].
So, which is right? Probably is it time to look more into how we perceive abstract objects and apply laws of transformation on them! The first time is most often bound to fail since our nerves are not used to the 'constraints at infinity' or to the 'laws of convergence/divergence'. It is only by careful analysis and playing quite a bit with dilemmas, we could come to a logical conclusion about something that is a relative of infinity. Infinity is just slightly different from being in finity.
