Galileo's Paradox of the Infinite

The modern story of the infinite begins with Galileo. Galileo discovered something interesting about unbounded sets. He discovered that if you assigned a mapping between two such sets (in this case I will look at the sets of natural and positive even numbers) you will not run out of elements from the first set before running out of elements from the second. Mathematicians refer to this as a one-to-one (1-1) mapping. Here is the mapping between the natural numbers and the even numbers:

     1   2   3   4   5   6   7 ... n
     2   4   6   8  10  12  14 ... 2*n

This 1-1 mapping seems to imply that the set of natural numbers is the same size as the set of even numbers even though the set of natural numbers is a proper subset of the even numbers.

Galileo refered to this as a paradox. The Hungarian Mathematician Bernard Balzano (1781-1848) was interested in the nature of space and time. His thought that this paradox was the fundamental nature of the infinite. Note, many texts today define an infinite set as a set that can be placed in a 1-1 correspondence with a proper subset of itself.

Code to test the difference between the natural and direct mappings.

// The Direct Mapping:
int i = 1000; // test size
print("Natural Mapping");
int evens = 0;
for (i=1; i<n; i++) {
  println(2*i);
  evens ++;
}
println(evens + " evens");

// natural mapping
// % is the modulo function
// it returns 0 if i% is even

evens = 0;
println("Natural Mapping");
for (i=1; i<n; i++) {
  if (i%2 == 0) {
    println(i);
    evens++;
  }
}
println(evens + " evens");

I hold the traditional view that Galileo's paradox is a true paradox. The reason the paradox exists is that there are two possible ways to create the set of even numbers from the set of natural numbers. For sake of demonstration, I will call these the direct and natural approaches. In the direct approach, you multiply each natural number by 2. In the natural approach, you simply remove the odds. The difference between these methods is easy to see when working with finite sets.

Multiplying the first n natural numbers by 2 produces a set of evens. This set created by a direct mapping will have n members—the same size as our original set. The largest member of the set is 2*n. Please note that half of the numbers in our new set are greater than or equal to n.

The set created by dropping the odds will have n/2 or (n-1)/2 members (depending on if the value of n is even or odd). The largest member of the new set is either n or (n-1). No members of the set created with a natural mapping is greater than n.

The two methods create different result sets with different characteristics.

The Set of Halves

The set of halves is a little bit different from the set of evens, as the integers are a subset of the halves. The direct mapping (dividing by 2) would look as follows:

     1   2   3   4   5   ...
    1/2  1  3/2  2  5/2  ...

The direct mapping produces a set that is the same size as the natural numbers, but the largest member has the value of n/2. As the integers are a proper subset of the halves, the natural mapping is reverse of that for the evens:

    1/2  1  3/2  2  5/2 ...
         1       2      ...

The size of the integers is just about half the size as the set of halves. The largest integers is trunc(n/2), where trunc returns the greatest integer less than a given value.

Infinity gets complicated as there can be more than one way to map a set onto another set. Each mapping could have different attributes.

Wave Particle Duality

When looking at finite sets, it is clear that the method used to create the set affects the characteristics of the set. Looking at infinite sets, the distinction becomes muddy; hence, there is a paradox. If I were to say: "take the set of all even numbers," I create a paradox because I have not specified which form of mapping I used to create the set.

Intuitively, when set B is a subset of A, I tend to think of natural mapping. People who are told to approach the theory by thinking first of 1-1 functions are likely to conisder the direct mapping as the proper mapping.

One of my disagreements with the current method of discussing the infinite is that transfinite theorists tend to reject the existence of the natural mapping, and hold that direct mapping is as a higher level of thinking than the petty natural mapping.

I contend the that Galileo's paradox is a true paradox, and that there is not an easy resolution for the paradox since both mappings are valid. Furthermore, I contend that infinite sets will behave much like the infamous wave/particle duality of light. Tests to prove light is wave, prove that light is a wave. Tests that prove light is a particle prove that it is a particle. Tests with built the assumption that the set of even number was created with a direct mapping will prove it such. Those built with the assumption of a natural mapping will prove its point.

It is physically impossible to create an infinite set. We can only explore the concept of infinity with finite tools that examine the subject. In creating such tools, we must be careful not to confuse the attributes of our tools with the attributes of the subject we wish to study. Limiting ourselves to 1-1 functions will impoverish our view of the infinite as the 1-1 function carries a hidden assumption of the direct mapping. When given just the set of evens, we do not know the mechanism used to create the set. The tools we use on the set will impose their point of view. Subtle changes in the way we approach an infinite sequence might have dramatic and unexpected results.

Infinite sets seem to be affected by even small changes in the way they are constructed. Lets look briefly at summation of the series [1, -1, 1, -1, 1, ...]. This series repeats the pattern 1, -1 indefinately. Depending on how we perform the addition, we get different answers. Attacking the problem from left to right gives me an answer that fluctates between 0 and 1. Grouping the problems using the commutative property of addition gives me different results depending on how I group the numbers:

     (1 + -1) + (1 + -1) + (1 + -1) = 0 + 0 + 0 + 0 ... = 0
     1 + (-1 + 1) + (-1 + 1) + (-1 + 1) = 1 + 0 + 0 + 0 = 1

The commutative property does not seem to hold for infinite sets. When we perform long division on 1/(1 + x) and 1/(x + 1) we get different results as well. The following two demonstrations show the long division for the two equations. Although they are essentially the same equation, when expanded to infinite terms they give different results. (I will use the carat "^" for the exponential. So x^-1 = 1/x):

              1 - x + x^2 - x^3 ...
    1 + x ) 1
            1  +  x
                 -x 
                 -x - x^2
                      x^2
                      x^2 - x^3
                           -x^3 ...

                x^-1 - x^-2 + x^-3 ...
    x + 1 ) 1
            1 + x^-1
               -x^-1 
               -x^-1 - x^-2
                       x^-2
                       x^-2 + x^-3
                              x^-3 ...

Simply changing the way we create the infinite expansion affects the result of the expansion. Setting x = 2 we see the infinite expansion of 1/(1 + x) creates the infinite expansion (1 - 2 + 4 - 8 + ...). This sequence diverges. While 1/(x + 1) = (1/2 - 1/4 + 1/8 - 1/16 ) converges to 1/3.

I was told in grade school that 1/(x + 1) = 1/(1 + x); However, when I use the different functions as the base for an infinite expansion, I get radically different results. One diverges for x = 2, the other converges. Oddly enough the series that diverges for x = 2 converges for x = 1/2. Regardless, the value of infinite sequences appear to be affected by the way we create them. That means it is difficult to distinguish a property of infinity from the tools used to investigate infinity.

The infinite hotel example is often given as proof that n = n + 1 for n = ∞. However, as soon as you try to look at the construction of the infinite hotel, you will find that there really is no way to move n + 1 things into n slots. (Ref: infinite hotel)

Just as the liar's paradox is a true paradox, Galileo's paradox of infinity is a true paradox. Unfortunately, Galileo's conversation between the wise Renaissance man Salviati and the dumb as a stump Simplicio was to feed the the notion that there was a higher sort of thinker who was able to comprehend infinity in its totality. The 20th and 21st centuries were full of just such creatures who felt that they were were on the dawn of a new transcendental consciousness, and that culture was getting ready to digi-volve into the next level of being.

As infinite sets all share the common attribute of being unbouded. By the time Georg Cantor appeared on the scene, a large number of people had taken to the opinion that all infinite sets were the same size. Even worse, many concluded that the natural mapping was simply a fault of failed human intuition.

German intellectuals following Kant's "transcendental idealism," were eager to find ways to refute common sense and intuition in favor of a new logic from a new genre of philosopher kings. So while mankind was taking great leaps forward with the new scientific method and other derivatives of Aristotelian logic. Galileo's paradox was to lead to a great step backwards when twisted by dialecticians of the German idealist tradition.

From here you have two options. If you would like to can go straight to the refutation of the diagonal method, or learn about how the diagonal method was one of a dozen flawed theories based on opositional logic. The main thrust of the next article on dialectics is simply to discuss the different oppositional theories that sprouted in Europe and that led to the great death camps of Germany and Russia. While the oppositional theories appeared to create a paradise for some, they created a hell for humanity as a whole.

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