Mathematics : How far is infinity?

As a kid and even as a grownups, we wonder about many things. Science is a discipline that is bent on satisfying man's curiosity. Just as my friend believed he could lift a glass of water with his mind after watching the movie series, Merlin, I also believed that infinity can be reached.

^{The symbol of infinity. credit: wikimedia. Creative commons license. Contributed by the US Department of Veteran's Affairs}

The concept of infinity came to me as a kid when I went to have a haircut, the barbing saloon had two mirrors facing each other, one meant to be used by the person who is having a haircut and another hanging on the wall just opposite the first one. As I was having a haircut one day, I decided to checkout the back of my hair by looking at the mirror hanging just a little distance from my back from the mirror situated before me.

I got more than I bargained for cause I saw both my face and back of my head many times with each view becoming smaller and smaller. I became interested in what I saw and when I went home, I brought two smaller mirrors, placed their shiny sides to face each other and leaving small space between them. I placed a pencil in between the two mirrors and I got the same effect as before; same image appearing countless times with each preceding image becoming smaller than the image before it. This was my first encounter with infinity.

Infinity is a very common word, both in science and in mathematics (especially in calculus) and simply means limitless size and or length. As kids we all learnt how to count the natural numbers which is usually in the sequence of 1, 2, 3, 4, and so on but up till now, we're still counting. Mathematics simply described Real numbers as a set consisting of positive infinity and negative infinity making the set of a natural numbers a subset of real numbers.

Irrespective of how go in the infinity, we would still find out there's more. The new internet protocol version 6 contains addresses that have the size of 2 to the power of 128, our solar system contains stars with the number of about 10 to the power of 23, the universe contains atoms with number up to 10 to the power of 87 while scientists estimated the lifespan of the universe to be about 100 to the power of 100, yet there's still more!

Lets consider a set of natural numbers consisting of only the even natural numbers which is usually of the form {2, 4, 6, 8, 10, ....}, if we match this set of numbers to the set of natural number, we could say that the even natural number has size that is twice that of the natural numbers but this is wrong. This is because for every value of the natural number, there exist a positive even number that can match the natural numbers as illustrated below.

^{designed by me @yandot using windows paint}

But this is not the same for all instances of infinity cause there are some infinity that are actually bigger. This brings us to the two types of and as you might have already guessed it, they are:

Uncountable infinity and

Countable infinity

The set of natural numbers which was described earlier perfectly describes a countable infinity but we cannot say the same for the set of real numbers which the set of integers, natural numbers and fractions. Take for instance, the gap between 0 and 1, is actually an infinity! This is because whatever fraction you think exists between 0 and 1 can always get smaller and or bigger, hence you can add as many zeros as possible in the decimal point between 0 and 1 inexhaustibly. Hence the number 0.000000000000001 and 0.9999999999999 exists between 0 and 1 and can get bigger without equating to a 1 or get smaller without equating to a zero. This is of the type uncountable infinity.

Irrespective of the fact that both type of infinity cannot be attained, the countable infinity is lesser both in size and length when compared to the uncountable infinity. As an example, the number existing between 0 and 1 can be said to be as the whole natural numbers because we can match the whole natural number with a number between zero and one!

Phenomenon Of Infinity in Reality

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The concept of infinity as we have seen is very weird and at this point, I want us to see infinity in its full manifestation.

1. The Paradox of Omnipotent Being

They say that religion is the opium of the masses and I dare any christian to come forward and debunk this claim. This very paradox comes in the form of a question and the question is, is it possible for an almighty being with a limitless or infinite power create something so big and mighty that he, the creator cannot carry it?

^{image source: wikimedia. Creative commons license. Contributed by Shaahin}

For instance an omnipotent god who can infinitely do "all" things, should also be able to create something he cannot displace. If he creates something he cannot displace or move, then he is not infinitely powerful after all.

2. Banach-tarski Paradox

When I was in the high school, the seniors (bullies) do give us $5 to buy $10 worth of edibles, funny right? This is very similar to what Banach-tarski paradox has to say. It is a theorem and it states that it is possible to obtain two balls from a single ball with the resulting two balls having the same geometric properties as the original ball.

^{image source: wikimedia. Creative commons license. Contributed by Benjamin D. Esham}

This very complicated theorem bases its claim from the fact that a ball which has spherical shape contains an unlimited amount of points in

3. Zeno's Paradox

The popular saying that a journey of one thousand miles begins with a step but what about a journey of one step? Zeno of Elea who is a Greek philosopher who believes that motion is nothing but an illusion because for a specified distance to be covered, an infinite amount of distance needs to be covered first.

To understand this paradox, suppose Mr. Blockchain is told to cover a distance of say 10 meters, obviously he needs to cross half of the distance which is 5 meters before reaching the said destination and before he reaches half of the distance, he needs to reach half of half of the distance (2.5 meters) and before he reaches half of half of the distance, he needs to cross half of half of half of the distance (1.25 meters) and the sequence continues like this infinitely.

^{image source: wikimedia. CC4.0 license. Contributed by Martin Grandjean}

This simply implies that for one to move from one place to the order he needs to complete an infinite number of tasks and this, Mr. Zeno believes is impossible. This is because apart from the initial problem of completing infinite task, the said movement contains no initial distance since any given distance can be continually divided into two and hence contains no initial point. The implication of this is that the motion cannot even begin in the first place.
Conclusively, Zeno’s paradox implies that any movement over a stipulated distance can neither be started nor finished, hence, motion is nothing but an illusion. Hence to move from one place to another, no motion is required in the first place!

4. The Hyperwebster Dictionary

The paradoxes that proves how boundless infinity is are many but I’ll stop in this one. Have you ever imagined a dictionary that is uncountable? Have you also imagined a dictionary that contains all possible words obtainable using the English alphabets? Well, I present to you, the Hyperwebster dictionary.

The idea of Hyperwebster dictionary was put forth by Dr. Stewart Ian in a bid to describe how boundless the points on a real number line are. The idea of the Hyperwebster dictionary is to combine the letters A to Z in every possible way giving rise to all English word both said and unsaid with no definite length.

The Hyperwebster dictionary of course contains both meaningful words and meaningless words. Words like SRRWTBQ, WTRIPVMXQESTYT, QUIDPROQUO, HELLO, WHAT, HOWAREYOU, WHATISYOURNAME, etc. Because of the fact that some letter combinations has no meaning, the company that published the uncountable dictionary decided to do so without associating a meaning to any combination. Hence, in the Hyperwebster dictionary, words are written in the form A, AA, AAA, AAAA, AAAAA, AAAAAA, …, AB, AAB, AAAB, AAAAB, …, ABA, ABAA, ABAAA, ABAAAA, …, B, BB, BBB, BBBB, BBBBB, …, and continues till, ZZZZZZZZZZZZZZZ…..

With this, the best way to punish a troublesome kid is to give him the hyperwebster dictionary and tell him to find the word, Abracadabra, lol.

Summary

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The concept of infinity has been a very old and fascinating one. From the symbol of infinity, it could be noted that infinity has no open ends, hence infinity describes things that are boundless. This concept is applied vastly in the field of mathematics and physics and many philosophers have put forward some paradoxes to explain the concept of infinity and some were described in this post in a very subtle manner.

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Thanks for your time.

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References

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1. Infinity ~wikipedia
2. Mackie & The Paradox of Omnipotence ~withalliamgod
3. Banach-Tarski Paradox ~mathfunfact
4. Zeno's Paradox ~wikipedia
5. Hyperwebster - an uncountable dictionary ~ravi-bhide