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Infinity - Does it exist? Can we Prove it? What about God?

By Miroslav Bonchev Bonchev Botev
When I was around 15 years old, I had a remarkable private teacher in mathematics by the name of Ionko Negencov. I usually had two or three lessons a week with him, and in some weeks even four or five. I didn’t need that much tuition, but I really enjoyed the lessons. I did an awful a lot of work in each session and I loved them. They were challenging, but all the effort I put in was repaid with the gratification of newly-gained understanding. I kept going to this extraordinary teacher for four years, even when I didn’t need any further maths tuition. That was the thing, I learned so much more than just maths from him. He helped was to discover the beauty of understanding.

At about the same time, I had two small mirrors in my room. One of the mirrors had a patch of corroded reflective surface in the middle. One day I realized that if I scratched a small hole in the corroded patch, I could see infinity by setting the two mirrors parallel against each other and looking through the scratch. After all, I needed to see infinity as I was studying limits and derivatives for first time. Indeed it worked, but did I really see infinity? What exactly did I see? In my 15 year-old mind, I was convinced that infinity did exist, and not only had I seen it, but I could even create it anytime I felt like. I was so fascinated with this small "discovery" that I would stare into "infinity" through the mirrors several times a day for the next 2 or 3 years. But was that indeed infinity?

Years later, after I obtained my MSc in Computer Science, and had gained a lot of experience designing and writing complex software systems, I started a second degree, An MSci in Pure Mathematics. While studding mathematics, I came to realize that my empirical proof of infinity was not a proof at all, simply because I was assuming the existence of infinity already. During an algebra lecture in the first year of my mathematics degree, a lecturer gave a mixture of definition and explanation of infinity. Although I believed in infinity (having my proof) I found his explanation to be quite flaky, so after the lecture I went asked him about it, believing that he would have something more substantial under his belt. Very disappointingly, he didn’t. The typical explanation of infinity is something along the lines of: "you can always count one more". Well, this is simply silly! For a computer scientist there are many questions provoked by this "satisfactory" definition of infinity. For example, "Who is doing the counting?", "for how long?", "By what algorithms?", "Where is the space where the objects are held?", "What is that space?", "How big is it?" etc. The more I questioned my lecturers and professors about infinity, the more I realized that they did not have anything better than the unsatisfactory explanation of "you can always count one more". Because one can count 1+1=2 it does not necessarily mean that 11+1 can be also counted and does exist. If the nature of numbers is the same as the nature of computer registers, then they finally will wrap up and start from the beginning. Therefore, without knowing the nature of numbers, we cannot say at all that we can always add one number. It may be the case that numbers grow infinitely, or that they wrap up after some biggest number, or that they reach a certain "highest number" and remain at it regardless how many ones are added to it – like a coil or capacitor. Numbers may have other behaviors/natures, but without knowing it, we cannot say such a thing as: "you can always count one more" at all. It seems to me that numbers exist up to a maximum number for every context. For example, if one talks about the bricks in a building, then the maximum number is some value related to the number of bricks in said building. But if one is to talk about the number of atoms in a single brick, the maximum number changes to another value that is related to the number of atoms in that particular brick. In other words, it seems to me that numbers are defined by the context in which they are used. However, the objective of this article is to give you two proofs that infinity does not exist, so any further discussion about the nature of numbers is outside of its remit.

In my search for the truth on the matter of infinity, I made enquiries to multiple mathematicians, only to receive further unsatisfactory answers. Professor Cameron, a renowned mathematician professor of mine who was the leading figure in the infinity documentary by BBC in 2010, told me that: "If a mathematician can think of infinity then it exists." Meaning that the fact that a mathematician can think of infinity is sufficient to justify the belief that is exists. Well, I respectfully disagree. One can imagine rolling a square-circle down the street, or why not water running up a hill. I need a solid demonstration that infinity really exists, not hand-waving, intimidation, or let us agree in order to be friends. The "definitions", "proofs" and "explanations" of infinity that I have seen are always plainly fallacious, either contradictory or circular, or assume that infinity already exists in some disguised form. For example, in this article, in order to justify their conclusion that infinity exists they (1) omit information (remove the context) and (2) assume that it exists. This is the typical "you can always count one more" - well NO, I cannot add another spoon of water to a cup which is already full. It overflows. In Wikipedia, they try to intimidate the reader with "The greatest minds believed in infinity..." adding seemingly complex nonsense to it, and a glorification of Cantor (as usual). Besides the usual removal of context, they assume that infinity already exists by assuming that every length can be divided (as usual), and by assuming that (a) { 1, 1, ..., 1, infinitely many times, 1 } is the same as (b) { 1 }. In (a), once again they assume that infinity exists; secondly, it is quite obvious that either (a) is not well defined or (b) is NOT (a).

Since I could not find mathematician or a book that gave a satisfactory definition or clear proof that infinity does exist, I sat and proved that it in fact infinity does not exist. Here I will give two proofs. They are simple, but it may help if one has a computer science background and experience in defining consistent types.
Theorem: Infinity does not exist.
Proof I
Premise: Everything in the world is uniquely identifiable, in other words if two objects have the same identifier in some complete identification scheme, then it is the same object.
Therefore infinity does not exist, for 5 is not 6, but 5 + ∞ is the same as 6 + ∞ is the same as ∞. Where 5 and 6 are some objects able to relate to infinity via some operation called "+". So either infinity does not exist or no such operation exists. If infinity does not exist, we are done. Suppose infinity does exists and no such operation "+" exists. But if no operation on infinity exists, then we cannot even refer to infinity, as reference is also an operation, but if infinity cannot be referenced then it does not exists. QED
Proof II
Every object has at least two properties:
  1. It lives in a space, i.e. it has a location (locality), i.e. every object is localizable.
  2. It has a type, i.e. specification.
But infinity:
  • has no location, for if it had a location we could point it out, and we cannot; and
  • has no type, for infinity has no specification.
(If objected, one needs to demonstrate a definition for infinity, that is consistent and can be instantiated at will, subject to appropriate (well-defined) but existent conditions and circumstances.)
So infinity is not an object. But, if infinity is not an object, then it must be a type, but types live in metaspace, so types are objects in the metaspace. However, infinity is not an object, so infinity does not exist since its instantiation (objectification) is never reached, always regressing into an upper and upper metaspace. So the definition of infinity is, at best, by infinity, but this is a circular definition and so it is invalid. So infinity does not exist. QED
Remark: For the reader who is not familiar with computer science, objects are instances of types defined in a metaspace, where the types are themselves objects.
Even God is not infinite. He is Complete, and that completeness may be incomprehensible to us, but that does not make him infinite. Note that nowhere in the scripture does it say that God is infinite. If He was infinite He would not be all-accounting and all-knowing, since infinity necessarily implies loss of information. The scripture says that God is Greater than the Heavens and the Earth, and that man cannot understand His ways, but this by no means implies that He is infinite. The notion that God is infinite was invented by theologians, or perhaps merely translated from paganism, as in fact one could trace the idea of infinity to ancient pagan Egypt!
So what did I see when I as 15? Well I was indulging in staring at "infinity" with my mirrors experiment, but all that I was seeing were only nested reflections, obviously they were many, but very far from infinitely many. The "infinity" I was observing in the mirrors breaks at the level of atoms when the smallest possible reflection is onto one atom. Obviously I was aware of that, but I believed that it was infinity and was closing my eyes to that fact only to maintain my belief. I was guilty of following a principle so eloquently expressed by Edmund Spencer: "There is a principle which is a bar against all information, which is proof against all argument, and which cannot fail to keep man in everlasting ignorance. That principle is condemnation before investigation."
In conclusion, the two proofs outlined above clearly demonstrate that infinity does not exist. This means that the world is much easier to understand and simpler to describe, and that awful a lot of theories are meaningless since they require infinity. For example: Real numbers ℜ existence is no longer justified, continuum does not exist, analysis becomes simpler, topology goes in the bin, group theory remains in the set of finite groups only, and the biggest nonsense of them all, Set Theory, is debunked. It is pity, but I don’t think that the fact that infinity does not exist will soon be taught in schools since there are too many people (mathematicians, philosophers, theologians etc) who need infinity to continue to teach their beliefs and theories as if they were true. Removing infinity will put many people from the above groups out of a job. Since these very same people are the ones who "decide" if infinity exists, considering human nature, it is likely that most of them will strive to keep the religion of infinity alive than go back to the student desk to acquire new skills and seek out other employment opportunities. The case of infinity is just one of many that show public schools and schooling to be an indoctrination schema as opposed to true education, which is damaging to pupils. They must be abolished so that private education (as opposed to public schooling) could take its rightful place in society.
Miroslav Bonchev Bonchev Botev
3-rd March 2011
London, England
To learn more about public schooling, please view this talk by the renowned educator John Taylor Gatto: "Beyond Schooling"
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