HOW TO COUNT TO INFINITY: IT BECOMES WEIRD!

Counting infinities has been a pain in the butt for generations of philosophers, theologians, mathematicians and of course physicists. How long can infinity go on being infinite? And how can we count to infinity is a profound question in the world of mathematics. So, let's start counting 1,2,3,....44,45,.....1323436477548578,.....(don't read the large numbers!)

• What is infinity? How did we stumble upon it?

     Infinity is not an ordinary number that can be counted, but a concept of something being unbounded and endless forever. The early Greeks gave the term ''apeiron'' which meant something not being finite. Zeno of Elea, Aristotle, Anaximander of ancient times and Issac Newton, Leibniz, Georg Cantor and David Hilbert of the modern period all tried their hand at taming infinity but in vain. The symbol ∞  was termed, ''LEMNISCATE'', in 1657 by John Wallis.                                                                    

     Nobody knows for sure who exactly came up with the idea of infinity. But for arguments sake maybe infinity is a remote correlation between man and an universal creator. That's debatable and I'm way out of my pay scale to discuss it here. Sorry.

• How to count infinities?(with paradoxical formulations)

     A German Mathematician, Georg Cantor wondered about the true nature of infinity. His wonder was the brainchild of a new branch of mathematics viz. Set Theory. He reasoned that numbers like 1,2,3...,99,100,....546,547,.. or -5,-6,..-15,-16,.....make up an infinite set that can be counted(listed). Decimals, Fractions like 0.21245579625,0.21245579626, 0.2124557627,...make an infinite set that is uncountable(impossible to be listed). Irrationals like √2=1.414213562.......goes on indefinitely without bound. Cantor basically found different types of infinity, and said ''some infinities are bigger than other infinities'' resulting in a paradox. So what did Cantor mean by his statement?

       An Illustration: Suppose there is a number line 1,2,3,..and going on up to infinity. Nobody knows for sure how many numbers are between 1 and infinity. But it is certain that there are more decimals between 0 and 1, than integers between 1 and infinity. The number line between zero and one goes like ....,0.012,0.013,0.014,0.015,... and likewise. Now if we assign integers from 1 to infinity with decimals between 0 and 1, the set of 1 to infinity would be a smaller infinity-set compared to 0 and 1 infinity-set. In other terms the number density between 0 and 1 is incomprehensibly higher than that between 1 and infinity. 

       Thus from here we know about listable and non-listable infinities. If I can retain the sanity of my brain then I could list all the numbers between 1 and infinity, no matter how much ink & paper it may take. But I cannot list all the decimals between 0 and 1, as there would always be some decimal beyond my list. This is called Cantor's Diagonal Argument, which I prefer to leave in peace with Cantor.

  Infinitesimal Calculus: We all try to survive our calculus course in school. Luckily I did with the blessing of Newton and Leibniz, the founders of infinitesimal calculus. Going down memory lane, dx is an infinitesimal change in the position of some particle at any instant of time dt. To find the instantaneous velocity of the particle we take dv=dx/dt . This is to say that dx is very small compared to x(position of the particle) but if we add up all the dx then we have x. I remember my teacher saying, "don't question and integrate this ∫f(x)dx from -∞ to +∞ '', for which I hated him. I couldn't figure out, how come we integrate a function with an undefined limit. But later I figured out infinitesimal calculus; which takes into consideration all the small quantities to get a larger one. An infinite amount of infinitesimals constitute a finite quantity, be it a number, value or anything.

     Thus infinity cannot be counted because it is a concept of limit, as in some limit f(x), x tends to infinity. The only way of defining infinity is the use of set theory. Here are a few examples; the set of prime numbers, even numbers, odd numbers, squares of all even numbers and blah, blah, blah...

• Paradoxes of infinity?(mamma mia!)

I won't give more than two examples for the ease of my readers and myself.

1. Zeno's paradox of Achilles and the tortoise:  If the tortoise is given a head start, then no matter how fast Achilles can run he could never ever outrun the tortoise. It goes like this......

• The tortoise is at #1 and Achilles is at #0
• The tortoise is at #2 and Achilles is at #1
• The tortoise is at #3 and Achilles is at #2
• The tortoise is at #4 and Achilles is at #3
• And so on................

This means whenever Achilles reaches the tortoise's former position in some amount of time, it goes forward by another small distance in that time. Now Achilles covers the added small distance and the tortoise again advances a little forward, in a much smaller amount of time. Again if Achilles covers the new distance, the tortoise moves a much smaller distance ahead. Precisely for every infinitesimal change in the position of Achilles the tortoise leads by another infinitesimal change in position. Thus Achilles needs an infinite amount of time to cover all the infinitesimal distances and unfortunately he never wins the race.

Race between Achilles and Tortoise. (created with powerpoint)

   

     2. Hilbert's infinite hotel paradox: Suppose there's a hotel with infinite number of rooms and an infinite boarders. If a new boarder comes, the hotelier tells every boarder to shift their rooms by one. So that, ∞+1= . Now if an infinite number of new boarders come then also the hotelier tells every previous boarder to shift to a room twice of their own. So a person on room number 25479061 goes to 50958122. By this way there is always a vacancy in the hotel no matter how many boarders come and ∞+∞=∞ . I'm giving a very little amount of formulas here and i really beg my readers not to be angry at me. Here they are:∞+1=∞, ∞-1=∞, ∞x∞=∞, ∞/∞=?, ∞x0=?, ∞/0=?

• Are space and time infinite?(oopsie!)

     This is just bad. Coz a few decades ago it was still believed that the universe is eternal. The steady state theory voted for a universe where matter and energy are continuously created to maintain an uniform density throughout the universe. There also came a theory of an oscillating universe which expands and contracts with the passage of time. Many scientists are still in favor of a steady state model. But the discovery of the cosmic microwave background radiation (CMBR) hinted at some big bang- a beginning of the universe. So, a lot of unanswered questions came into light-whether the universe has a beginning or if it will keep on expanding forever?
     How did the universe begin? How long will it keep on expanding? Is the universe infinite? Suppose we can go to a timescale 10^10^10^56 years into the future which seems infinite apparently but we can't go to the infinite past right? If we try to go back to the big bang that happened 14 billion yrs ago, we are left with nothing but a big bang singularity.
     Now the gravest thing happens. The naked singularity at the beginning of time forbade us to see beyond the beginning of time itself. Nobody knows for sure how that singularity came into existence and if there's a "before'' before the big bang.
     Zeno of Elea argued space and time to be infinitely divisible, which made space-time to be non existent. The arrow of time froze, which made motion impossible. But we see space to be expanding indefinitely and within a trillion years from now, the cosmic light horizon becomes vast enough to make the big bang undetectable with the present technology. At that point in cosmic evolution, space and time would seem to be infinite from our earthly perspective.
     Physics would be going crazy at 1 trillion years from now. And so am I. Thus I find it wise to leave it here and implore my readers to tell me of their thoughts.