S.K.Venkatesan
We show here that just like the resolution of Zeno’s paradox, contradiction between countability and uncountability can be resolved. Here it is shown that contradictions don’t exist in an ideal world, and that they only arise due to elements in physical reality that challenge idealism. Contradictions gives rise to infinite hierarchies of classes and that these classes are distinguished better by scaling dimensions (Hausdorff measures), providing a new way to look at continuum hypothesis. We also show that the quantitative aspects of the problem of algorithmic complexity classes throws further light on these contradictions.
“Reason has always existed, but not always in a reasonable form” – Karl Marx
Contradictions are mills through which reality ebbs and flows. The right and left hand side exists as opposing manifestations. The fact that the perfect symmetry of the left and right is broken is what establishes their distinction. If Earth is a perfect sphere, then there would not be rivers and valleys that brings life to it. However, since the radius of the earth is almost a constant (to more than an accuracy of 99%), we are qualified to call it a sphere.
Contradictions are also a great source of changes to society and transitions from one form to another. Goliath was powerful when combat was direct physical conflict with hand-held weapons. When the giant Goliath calls David to come near him to fight him it is clear that only at that range that he can overcome David. David at the same time keeps his distance to deny him that opportunity and defeats him with his long-distance weapon. Inside the atomic nucleus such a conflict between the Goliath nuclear forces that operate only short distances release enormous amount of energy when it comes in conflict with the electrical repulsion that operates at long distances as the size of the nucleus becomes sufficiently large, as in the case of Uranium or Plutonium nucleus, when it is bombarded with neutrons causes nuclear fission releasing enormous amount of energy in short time. At the other end of the spectrum when the gravitational forces crushes electrical repulsion in Hydrogen ions, the Goliath nuclear forces take over releasing enormous energy producing the energy of the sun.
In this article we first consider classical Zeno’s paradox, countable infinity and uncountable infinities of higher order. We also show how these Hegelian contradictions lead to an hierarchy of infinities. We then show that the so called uncountable numbers can in fact be counted as well, using a close parallel with Zeno’s paradox, resulting in a serious fissure in continuum hypothesis and Gödel’s incompleteness theorem as well.
We will begin with the classical Zeno’s paradox that is usually explained in terms of Achilles and the tortoise [Aristotle, 350 BCE].
Classical Zeno’s paradox is illustrated by the story of the Achilles and the Tortoise. The paradox follows from the argument that if the tortoise takes a lead, then each time Achilles reaches the place the tortoise was, the tortoise would have moved further away, and however fast the Achilles chases the tortoise, it can never catch the tortoise, as each time the tortoise would have moved further away while Achilles is trying to catch-up. We will illustrate this in a much simpler way using the concept of recursive decimals:
We know from this that there are infinite sequence of numbers,
Let us now consider the cardinality of a set as the number of elements in a set, for example, if
Now consider the set of all subsets of
We would like point out that this process can continued indefinitely, i.e., we can construct power-set of a power-set or, in plain language, that there can be a set of all subsets of the set of all subsets of a set etc., i.e.,
The cardinality of natural numbers is denoted by ℵ,
Now consider the cardinality of the set of all real numbers between 0 and 1, i.e.,
Just as the number of drops of water in a glass of water cannot be counted like a bunch of bananas, we intuitively know that the set of real numbers in [0,1] cannot be counted like natural numbers. We will prove this now. Let us represent all real numbers between 0 and 1 in terms of their binary representation in some counting order,
By a diagonalization process discovered by Cantor we can construct a real number,
where
leading to a contradiction. We would like to point out that this proof is quite similar to the proof used in Zeno’s paradox showing that the Hare never catches the Tortoise, so like Zeno’s paradox we will prove also its opposite in the next section. For now, in this section, however, we will assume that this proof is correct and proceed further to qualitative results, i.e.,
In order prove this consider the binary representation of real number in (0,1),
We now obtain the corresponding subset of natural numbers by considering all the index positions of "1" in the above binary representation, i.e.,
Similarly for every subset of natural numbers we can construct a real number in (0,1). This implies a bijective mapping between the power-set of natural numbers and the set (0,1), thus proving the result. QED
So finally we also have the result that we can construct an infinite hierarchies of infinities, i.e.,
Now, we will establish a new result, that contrary to classical mathematics, it is possible to count the real numbers in (0,1) in a orderly sequence. This establishes a contradiction between countability and uncountability. First, let us consider the binary representation of real numbers (0,1) using a tree-like representation (Figure 1).
Now let us count the binary numbers from top-down-left-right in Figure 1:
0.0, 0.1,
0.00, 0.01, 0.10, 0.11,
0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111,
.........
By using Cantor’s arguments of diagonalization we get that the number,
is not in this counting sequence. However, contrary to this argument, we know Achilles will eventually catch-up, i.e., when the depth of the tree (which is now 3) reaches 14 (= 2 + 4 + 8), then the last entry in the right-bottom will be the number 0.11111111111111. What about the number 0.11111... , i.e., the number 1.0. Will it ever be reached in counting as it is an infinite sequence? The question posed is similar to the Zeno’s paradox, as we assume that counting takes equal amount of time. What if we count them very fast in parallel. Assuming that we count the items of the
which is a finite value. Here the crucial point is whether
One might argue that the point at infinity is just one point and how come this creates a one-to-one mapping from the set of natural numbers to the continuum. The point at infinity is a single converging point in time, but in the space of counting zero-dimensional points it maps to
Similarly, if we consider multi-exponential trees, we obtain,
At a philosophical level, we can say that in the ideal Platonic world, there is only one infinity (one truth and one god). Paul Cohen [Chow, 2007] proved using the concept of forcing that a consistent set of axioms of set theory cannot prove or disprove continuum hypothesis. Abstract ideas do perform an useful function of unifying various disparate concrete objects but as observed by Arnold (1998) , they mostly do not yield any thing concrete by themselves.
In summary, we found that the idea of countability breaks down, so it cannot be used to classify levels of infinities and all the Hegelian infinities collapse to a single infinity in the Platonic world. However, the concept of continuum hypothesis still remains true in this new context where we will no more be counting bananas but instead weighing them. In the next section (Section 3) we will construct a counter example to continuum hypothesis using a new way of distinguishing infinities with Hausdorff measures or the concept of scaling dimensions.
Although here we are only discussing about numbers and mathematics of set theory, these results have much greater implication in computer science as there are corresponding categorical mapping at higher levels that maps these domains to similar problems there. At an abstract level decision problems can considered as the function mappings,
where
In order to make this article self-contained we briefly indicate how the halting problem is related to the Zeno’s paradox, but for more details about the halting problem and it’s relation to the Gödel’s incompleteness theorem, please refer to lucid account by Aaronson [2013]. The halting problem constructs the two legally disjoint categories, the finite and infinite, and finds a contradiction between them. Here we noticed in our earlier sections that, just as in Zeno’s paradox, the infinite of events is contained within a finite interval of time. In an ideal world it is possible to count an infinite of things in finite time and only by imposing some conditions that are borrowed from reality that we can establish such logically disjoint categories and then only a contradiction can be established.
Although it is recursion that is establishing contradiction between the finite and infinite, it is also recursion that is capable of producing the infinite by finite means. It is the disease that some times provides the cure, like in the case of inoculation that injects the disease causing agent itself with reduced virulence in order to fight the disease.
Additional elements from physical reality are required, in order to restore validity of Cantor’s proof on uncountability of the continuum [0,1] and also Gödel’s proof of incompleteness. In a previous section we established that the continuum can be counted, where the crucial step was that the
Just as in the case of Euclidean geometry, where one cannot prove that two parallel lines can never meet, these assumptions about infinities yield different categories of algebras. Of course, there are other simpler ways to distinguish discrete from continuum, i.e., by the fact that in the case of continuum, between any two distinct point in the set there is always another point distinct from the two that lies in between these two distinct points. We shall consider these techniques in the next section.
The continuum hypothesis states that there are no cardinal numbers between
Is there a set
defining the Cantor set to be,
In the above representation it seems as if the Cantor set can be represented as the disjoint union of countable sets. However, this is quite untrue, as the Cantor set is a totally disconnected nowhere dense set. So, it can only be written as the disjoint union of uncountable sets. This fallacy is due to the fact that countability of the finite-dimensional truncation of an uncountable set does not prove that the limiting set is countable, not that distinguishing between countable sets and uncountable sets is as easy as it was portrayed by classical mathematicians. Cantor set can also be symbolically written as a geometric set following the relation,
Or to put in simply, when we scale the Cantor set by 3, we get two Cantor sets, i.e.,
where
The points in the Cantor sets will not have digit "1" in them, i.e., say,
This may seem erroneous as
One then wonders how this is a counter example to the continuum hypothesis? The answer lies not in counting bananas but weighing bananas, as there is more geometric information in the Cantor set that is lost in counting rather than weighing them. It is this extra geometric information that is captured by the scaling dimension, which distinguishes it as a fractal object existing between 0-dimension and 1-dimension.
It also becomes apparent that the idea of countability is rather useless, as although the set of rational numbers are countable, they are dense in the set of real numbers and their effective scaling dimension is one and not zero.
The ideal way of mediating between the discrete and the continuum is by the use of Dirac distributions or the Heaviside function.
Let us now consider Heaviside function
Let us now define,
Now consider the support of the function,
Define,
There are many discrete sets that obey the property,
for them there is hope. However, for sets that do not satisfy this property we will only consider it's regular equivalent
In the case of the set of rational numbers we get,
But for Cantor set
It is possible to measure such sets with confidence.
Of course, there are pathological sets like the Vitali set,
which are not even well-defined; in the sense that there are multiple avatars of this set possible and it is possible to define them as subsets of
Of course, we invented all this mechanism to convince the audience that,
and so,
i.e., that they should be identified with
The method of identifying rational numbers of the form
One of the essence of uncountability is the fact that one cannot find the next number in the ordered list and this fact is satisfied by both the set of rational numbers,
In reality, mathematical idealizations are scale limits of real problems. Just as we cannot have a measuring scale of infinite length, these limiting problems are unrealistic abstractions, but they are useful to capture essence of some problems as a simplified cartoon sketch.
In our earlier sections we saw how the continuum hypothesis and set theory could be understood. In the same vein, let us consider a simple NP-complete problem, the subset sum problem. The subset sum problem is the problem of identifying a set of
Now let us consider the approximate version of this problem and see how it is solved. We will give a rough sketch here, but for more details please refer to the source.
We can pose this problem as:
Find numbers
such that
In the above approximate problem the introduction of an
then it truncates to a polynomial but the limit is an exponential.
In order to establish that it remains a polynomial in
We will now consider the simplest version of this problem:
Consider the binary representation of
$$M = \underbrace{100110011\ldots}{m \text{ bits}} = \sum{i=1}^{m} f_i 2^{i}$$
where
Now consider the set, for
where
Instead of this set, let us consider a special case of this set,
where
Let us now create a subset of natural numbers that has "1" in the $m-i$th-position of the binary representation of
The solution of the problem then reduces to checking if
which only requires an effort of the order of
You might wonder how can a given set of arbitrary numbers arrange themselves in exact powers of 2, an event of very low probability (probability is as low as
Consider the set,
We can now coarse-grain this as,
Thus we are able to quickly zoom into the solution through a tree-like sweep. Of course, we need estimate the complexity of these steeps in detail, as there could be strange distribution patterns that challenge our approach.
Subset sum problem is the simplest of the NP-class problems. Although this Subset sum problem seems quite different from the problem of prime decomposition problem, it can be transformed into a problem of product decomposition by mapping each number of the set into it's corresponding power of say 2, e.g.,
i.e., the Subset sum problem can be considered as special case of the Subset product problem where the sum is replaced by multiplication.
Of course, it is easy to verify whether the product of