The Elusive Infinity

I have rarely written articles on mathematics, but off late a few ideas have occurred to me which I would like to share with everyone. I have come across palindromes in languages. Take the word MALAYALAM for instance. It reads the same forwards and backwards. Similarly we have numbers that read forwards and backwards. These are called reversible numbers in mathematical jargon, but since I like the word palindrome better, I will call it palindrome for numbers also. Purists can think of it as palindrome_n, where n is to denote numbers.

Here are some palindromes of relevance to this discussion.

1, 22, 343, 43234, 5432345 and so on. By intuition we can see that if keep proceeding, the last palindrome would start with infinity and end with infinity.

Let us move towards developing a set of mathematical series to see where the last palindrome is hidden.

I will not go too deeply into how to come up with these series. For our discussion I am going to use the character ~ to denote infinity since I am unable to find a character for infinity in my computer key board. Here is the function that comes to my mind.

f(k, n, a, r) = a + (n – 1)r

Where a is always the first term of the series and all the other terms can be found using the second part of the function after the comma. And here are the few other assumptions.

k is the number of series and ranges from 1 to ~
n is the number of terms in the kth series and ranges from ~ to 1 and
a = k and r = k

Here I present the set of series that concern us derived using the above function as we move from k = 1 to ~. Since n decreases as k increases, the first term of each series would have ~ terms, the number of terms in the second series would be 1 less than ~ and so on till we reach the last series where k = ~ and there is only one term left which is also ~.

Here I present the series starting from the 1st and moving to infinity

1,2,3,4,........................................................................................~
2,4,6,8,.................................................................................(~ - 1)
3,6,9,..............................................................................(~ - 2)
4,8,.............................................................
............................................................
................
............
.........
......
...
~

By intuition, it is clear that our palindrome of infinite length which starts with ~ and ends with ~ is hidden somewhere in the above set of series. It can begin with the last term of the first series or the first term of the last series. In all the above series only positive numbers are present. So let us go ahead and define what we can call a positive space. From the above equations we arrive at a space where only positive numbers can exist and there is no place for 0 and negative numbers.

Similarly by extrapolation we can define what can be termed a negative space where all quantities are negative and there is no room for positive numbers or 0. So we have only -1, -2, -3, and so on up to -~. But the point is we are explicitly attaching a minus sign to the numbers. If negativity is to be assumed as an inseparable characteristic of the quantities such that the - sign can never be separated from the numbers then there is no difference between our positive space and the negative space.

So let us make that assumption and deal with positive spaces alone.

We have assumed that there is no place for zero or negative numbers in our positive space. So when we subtract any number from any number we have to get only a positive result. So

2 - 1 = 1
1 - 2 = 1

Since zero is not permitted in our space we have to do some thing about dealing with subtracting equal numbers. When we subtract equal numbers we cannot say it equal to zero. So let us leave that operation as undefined.

So

1 - 1 = undefined
3 - 3 = undefined

Let us in particular look at our palindrome stretching from ~ to ~. This can be likened to a straight line stretching endlessly in both directions.

Let us assume that the mid point of our line is X. And both ends are at ~.

So by our definition

     ~ - X = X - ~

     2X = 2~

Hence

     X = ~

Hence we arrive at the fact that the mid- point is also ~. Now here is a little problem we face. ~ to ~ does not make any sense. We all are all aware that going from one point to another makes sense if there is some difference in magnitude of the two quantities. So at this point  I try to rationalize a bit and try and understand what the ~ at mid-point actually means.  

The ~ at the mid-point can actually mean infinitesimally small. The magnitude if the smallness is equal to the magnitude of bigness.

Hence we have a situation where the beginning, end and the middle are all ~. The next question is how do we calculate the magnitude of ~ so that we understand it? That is the problem? Now here are a few of my thoughts.

Let us borrow a few things from our real world and look at our endless straight line so that we can at least try and understand it. Let the basics remain the same. Let us deal with the portion of the line that we recognize, and deal with it using real world ideas. Here is our straight line again so that we understand it.


   ~--------------------|---|---|---|---|---|---|---|---|-----------------------~
                                      4    3    2    1    ~    1    2    3    4

How is it possible to measure infinity using this? Let us convert this to a grid.

                                        4    3    4    1   ~     1    2    3    4
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~---------------------|---|---|---|---|---|---|---|---|---------------------~
   ~------------------                                                       ~
   ~------------------                                                       ~
   ~------------------                                                       ~
   ~------------------                                                       ~
   
And the lines end at infinity.

Now look at the two points 1 and 2. And look at the lines going downwards. We have joined 1 and 2 with a line or an artificial bridge of our own. My theory is this. If we move from 1 and go down the vertical line and arrive at 2 without building any artificial bridges of our own, then we would know whether we are dealing with a bottomless chasm or a genuine path that is real and exists. We would also know the magnitude of infinity.

With all humbleness I say that I will continue with this article if a mere mortal like me is able to find the answer.