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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 lessons. I didn’t need
that much tuition, but I really enjoyed the lessons. I did awful a lot of work at each lesson and I loved them –
they were challenging but all effort I put was repaid with gratification of understanding of something new and
of course with the due acknowledgments. I kept going to this extraordinary teacher for four years, even when I absolutely
didn’t need it. I didn’t learn only maths from him, but mostly to think for myself and stand for what is right,
in other words not to buy anything not worthwhile. I guess one of the things that I learned there was to be a grown man.
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 scratch a small whole in the corroded patch I could see the infinity by setting the
two mirrors against each other in parallel and looking through the scratch. After all I needed to see infinity as I was
studding limits and derivatives for first time. Indeed it worked, but did I really see infinity, and what exactly did I see?
In my 15 years old mind I was convinced that infinity does exists, and not only that I have seen it, but I could even
create it anytime I feel like it. I was so fascinated from this small "discovery" and from looking into the infinite that as
a matter of fact I was staring at it at least several times a day for the next 2-3 years. But was that indeed infinity?
Years later, after I graduated MSc in Computer Science, and after I had a lot of experience designing and writing complex
software systems I started a second degree this time in 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 has something more substantial under his belt. Very disappointingly it was his best. The
typical explanation of infinity is something along the lines: "you can always count one more". Well, this is simply silly!
For a computer scientist there are many questions provoked by this "satisfactory" for the common man "definition" of infinity.
For example "Who is doing the counting?", "for how long?", "By what algorithm(s)?", "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 do not have anything better than the unsatisfactory "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 exists. If the nature of numbers is the same as
the nature of computer registers, then they finally will wrap up and start from beginning. Therefore without knowing the nature of
numbers we cannot say at all that we can always add one and get a greater with one number. It may be the case that numbers
grow infinitely, or that they wrap up after some biggest number, or that they reach some biggest number and remain at it
regardless how many ones are added to it – like a coil or capacitor. Numbers may have other behaviour/nature but without
knowing it we cannot say any such thing as "you can always count one more" at all. It seems to me that numbers exist up
to maximum number for every context, that is specific maximum depending on the 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 the building. But if one is to
talk about the number of atoms in a single brick then the maximum number changes to another value that is related to number
of atoms in that particular brick. In other words it seems to me that numbers are defined by the context they are used in.
However, here I have the objective to give you two proofs that infinity does not exist, so I will discuss the nature of
numbers no further in this article.
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In my search for the truth on the matter I enquired about infinity form multiple mathematicians to always receive unsatisfactory answers.
Professor Cameron, a renown mathematician professor of mine who was the leading figure in the infinity documentary by BBC last year (2010)
answered to me that: "If a mathematician can think of infinity then it exist." 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 on a hill. I need a solid demonstration that infinity really exist,
not hand waving, intimidation or let us agree so we are 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
wiki 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; second 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 giving satisfactory
definition or clear proof that infinity does exists 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.
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| Theorem: Infinity does not exist. |
| Proof I |
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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. |
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Therefore infinity does not exists, 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
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| 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. |
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(If objected, one needs to demonstrate a definition for infinity, that is consistent and can be instantiated at will, subject of appropriate (well defined) but existent conditions and circumstances.) |
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So infinity is not an object. But if infinity is not an object it must be a type, but types live in metaspace, so types are objects in the metaspace. But infinity is not an object, so infinity does not exists, 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 is invalid. So infinity does not exist. QED |
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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.
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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 scripture says nowhere 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!
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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 infinity. 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 is infinity and was closing my eyes to that fact only to maintain my belief. I was guilty 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."
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| In conclusion, by the two proofs above infinity clearly 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 in fact meaningless since they require infinity. For
example: Real numbers ℜ existence is no longer justified, continuum do not exist, analysis becomes simpler, topology goes in the bin,
group theory remains in the set of finite groups only, not to mention the biggest of all nonsenses Set Theory, and so on and so on. It
is pity, but I don’t think that the fact that infinity does not exist will be soon taught in schools since there are too many people:
mathematicians, philosophers, theologians and related to them who need infinity to continue to teach their beliefs and theories as if
they are true. Removing infinity will result in the need for many people from the above groups to acquire new skills, and since these,
very same people are the ones who "decide" if infinity exists, considering human nature it is most likely that most of these people
will prefer to keep the religion of infinity alive than going back to the student desk to acquire new skills and seek for something
else to give them income. The case of infinity is only one of the many that demonstrate that public schools and schooling are an
indoctrination schema as opposed to true education, damaging to pupils and that they must be abolished so that private education
(as opposed to public schooling) could take its rightful place in society.
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Miroslav B. Bonchev
3rd March 2011
London, England |
| P.S. To learn more about public schooling please view this talk by the renown educator John Taylor Gatto:
JOHN TAYLOR GATTO: "BEYOND SCHOOLING"
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