EAP Teacher (English for Academic Purposes), Sabanci University

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## Does infinity exist?

I have been reading a lot about mathematics and astrophysics over my summer holidays. The question I have in my head right now is 'Does infinity exist?' or is it just a term to express what we cannot compute or understand yet?

• #### David Irvine

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Aug 29 2011: Interesting

To make a perfect circle you can draw parallel lines and keeping the distance between them, move them through a centre position all the degree's and parts of a degree. for the circle to be perfect, the number of parallel lines has to be infinite!

Problem,
There is no perfect circle, just a notion.
There is no such thing as parallel lines, just a notion

Now if it were possible we would have an infinity.

If we then transpose the circle to a sphere we have infinite infinities !

In the circle example we assume 2d - i.e. lines have no width If we move up a dimension to 3d then infinity becomes infinity to the power of infinity, now add more dimensions and it gets weird. This I think is a problem with some advanced physics, we use these notions and then take them way to far out of context. Maybe an example of proper chaos theory where our initial assumptions are very poor, and immediately we move to N-Dimension thinking like string theory.

Huge subject that I find fascinating, great question, I do think there is an answer, but I believe we do not have the language (spoken, written and mathematical) to provide that answer just yet. We can live in the excitement that there are answers and there are so many we can all help find them, if we just think enough and accept current failings readily.

Great time to be alive!
David
• #### Goran Malic

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Aug 30 2011: When extra dimensions are added infinity actually behaves very politely! In your example, adding another dimension to an infinite two-dimensional space does not produce an infinity to the power of infinity; instead, nothing really happens, infinity remains the same infinity you had in the beginning.

Think of it this way: a circle is a specific set of points contained in a plane, that is, it is a subset of the plane. A sphere is a specific subset of space. How many points are there in a plane? And how many in space? The answer is the same, both have an infinite number of points, so neither the circle nor the sphere can have more than an infinite number of points, i.e. they have exactly an infinite number of points, not infinity to the power of infinity.

Our intuition falls apart when dealing with infinity. A clear example is this one: consider the set of all positive integers 1, 2, 3, 4, 5 and so on. Now, cross out all the odd numbers in this set. You'll get the set consisting of all even integers 2, 4, 6, 8 and so on. The question is: which one is larger, the set of all positive integers or the set of both positive and even integers?

The usual answer is that the set of all positive integers is larger since we obtained the other set by crossing out odd numbers, but actually, they are of the same size, both are infinite. To clarify why this is so: well, we just rearrange the counting pattern of the even integers, we consider 2 as the first even integer, 4 as the second even integer, 6 as the third even integer and so on. In such a way we have established a correspondence in which 2 corresponds to 1, 4 corresponds to 2, 6 corresponds to 3, 8 corresponds to 4, 10 corresponds to 5 and so on, that is, to every even integer we can assign a corresponding integer. From this we conclude that there are as many even integers as there are all of the integers.
• #### Goran Malic

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Aug 26 2011: As it was pointed out earlier, infinity is not a number. More or less mathematically, it is a measure of size of the set of all integers. To be more precise, that is the definition of countable infinity in the sense that we know the successor and the predecessor of every integer, i.e. if we pick an integer "n", we can say that "n-1" is its predecessor and "n+1" is its successor and because of that, we can "count the integers".

As the name "countable infinity" suggests, there is, surprisingly, another kind of infinity called the uncountable infinity. That is the size of the set of all numbers and it is uncountable in the sense that we can no longer find the successor and the predecessor of any number. For example, if we choose 1 and try to find its successor, we could say that 1.00000001 is its successor - but that is not true because 1.000000001 or 1.0000000000000001 or 1.00000000000000000000000000001 (and so on) should be its successor.

As it turns out, the uncountable infinity is interestingly much larger than the countable infinity - although the term "larger" is not a good one; it should be substituted with "denser".

To answer the original question, as far as I know, there is no example of infinity in nature. Even the curved finite spaces such as the area of the Earth are finite in size, and one would eventually walk across every atom on the surface of the Earth.

As for the understanding of infinity, we cannot say that we understand it completely, but we can say that we have made significant progress ever since Georg Cantor inaugurated a branch of mathematics called "set theory" in the 1870s. There are still a lot of unanswered questions and some questions about infinity truly test the limits of mathematics as such - for example, the continuum hypothesis, which states that there is no infinity "more dense" than the countable infinity and "less dense" than the uncountable one, cannot be proved or disproved within the standard mathematical framework.
• #### Sharon Turner

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Aug 27 2011: Thank you Goran for this answer. You have given me some new leads in my study of infinity. I am going to read about Set theory and the continuum hypothesis.
• #### Nino Dundua

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Aug 23 2011: I think infinity exists only as a means of description, such as found in mathematics for example, or any other thing that exists only in the abstract. I do not believe that it has any real existence in the universe such as infinite mass or infinite size. The word 'infinity' is a descriptive term and not a measure of size, and I therefore do not see how it can be applied to anything 'real', as real things can be measured.

I have come across web sites, and maybe you have also, that claim that atoms can be subdivided down into infinity, and that they contain tiny universes within them, and no doubt tiny people as well. Although science has not yet been able to prove we have reached the ultimate elementary particle from which all complexity is built, there is very strong theoretical and experimental evidence to show that quarks could be it. Smaller than quarks enters the realm of energy, not particles, as in string theory. As matter has been subdivided down from complex objects, to parts of the whole, to molecules, to atoms, to particles, to quarks, at each stage we see a simpler model, each stage is less complex than the previous level. All of which is in perfect agreement with the Big Bang model that describes how all matter is built up from simple to more complex elements, stage by stage. So when breaking down complex objects into smaller parts, it would come as a bit of a surprise if suddenly an entire universe popped up at even smaller scales than wave energy. Entire universes tend to be a bit complex!

However, if string theory is shown to be correct, then tiny loops, or strings, of vibrating wave energy may be the smallest, but they are not particles anyway, and strictly speaking quarks aren't either, as they can not exist independently outside of a particle.
• #### Sharon Turner

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Aug 27 2011: Thank you Nina for this clear and comprehensive answer. I have also been reading about quarks etc. over the summer and smaller scales etc. Can everything real be measured though? How about pure numbers? (I think this is the term?!? They have no scale attached as yet. Do you think that we will produce a scale for them?
• #### Goran Malic

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Aug 28 2011: Numbers are an abstract concept and although they seem natural to us, they are not real phenomena, so the notion of size doesn't really apply to them in the same sense that the notion of sound or color doesn't apply to them (i.e. we cannot say what is the size of 17 in the same way that we cannot say what is the color or sound or smell of 39).

We can, however, discuss the size of a number relative to another number (i.e. 7 is larger than -4 or 0 is greater than -1 etc.) or its absolute value (i.e. its distance on the real line (the x-axis) from the origin) which is usually called the magnitude of a number.
• #### Christophe Cop

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Aug 23 2011: this conversation might shed some light:

http://www.ted.com/conversations/4380/beyond_just_science_and_religi.html
(http://www.ted.com/conversations/4380/beyond_just_science_and_religi.html?c=288429)

Now to the question: That greatly depends on what you understand by infinity...
(We suppose that) there is no infinite amount of matter or energy in this Universe, but we suppose infinite time in the future.
"infinity" also exists as a concept. As the number pi does exist... though asking whether pi exists (or any number),... you need to know to what it refers...

pi exists in circles, and infinity too...

(To conclude: the question is void)
• #### Sharon Turner

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Aug 27 2011: I cannot thank you enough for the reply and the links. Particularly the Khan Academy. What a revelation! This is what I have been searching for. Could you explain more about infinite time in the future. Thank you.
• #### Christophe Cop

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Aug 31 2011: We expect that the universe will not cease to exist, but evaporate into infinity (if I'm not mistaken), meaning that space-time will continue to exist in a near-empty universe.
• #### Farrukh Yakubov

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Aug 31 2011: Infinity is a function / property of things which we call infinite.
Infinity is 'never ending'ness of a process, field, or sequence.
• #### Nicoletta Strati

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Aug 30 2011: From my point of view infinity does exist! I can immagine infinity as an abstract space where the souls are traveling together to the eternity, another abstract term that wecould only understand by...exercise our 'third eye'...the strongest point of the human existance: the intuition!
• #### Sharon Turner

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Aug 23 2011: Dear Carlin,

I didn't think it was amateur. My maths knowledge is strange. I was weak at maths at school and quite honestly it terrified me until a year and a half ago when I realized that areas such as geometry, calculus and algebra made sense to me if I studied it by myself. There are gaps! As for Infinity, I am interested in the word and the concept within and beyond mathematics, from all areas. What does it mean to people? For example in your 1st comment you stated that outside of mathematics it is not useful. So is infinity for you only mathematical?

Sharon.)
• #### J Ali

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Aug 22 2011: Infinity as a number does not exist.....it is not a number... A number can be divided....infinity cannot... a number can have another added to it.. infinity cannot.. a number can have another decreased from it...infinity cannot...there is a difference between counting forever and counting to infinity....the former will never end and is possible...the latter is impossible due to the non-existence of infinity... infinity as a number is impossible because it is not one thing.....it is- if it is- a collection of things which are not infinite... i.e 1,2,3,4,5,6,7,.......so on... we look at things and then we give them a number....we look at a tree and we say one....then at another and we say two..and so on..numbers are in our mind..outside of our mind..there just is a thing..and another...we can keep counting for ever, but that still does not make a number infinite..just add another number to it... as for infinity as something that exists which has no limits in perfection...then I believe it exists....but that is another issue....
• #### Sharon Turner

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Aug 23 2011: A really interesting answer J Ali. Thank you. Yes, infinity would seem strange as a number or a quantity. But I am curious about your last sentence where you state that "infinity as something that exists which has no limits in perfection". Please explain more.)
• #### Chris Price

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Sep 3 2011: Could you divide infinity by infinity and come up with 1? 1dog divided by one dog = 1 dog? 10 divided by 10 = 1 ???
• #### Goran Malic

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Sep 3 2011: No, you couldn't for couple of reasons. The division operation is defined for numbers. Infinity is not a number. If you would try to work with infinity instead of numbers, you would get the following contradiction:

We say that a is divisible by b if there exists c such that a=b*c. If such c exists, it is unique and we say that c is the result of the division of a by b. For example, take a=7 and b=7. We have to find c such that 7=7*c. Obviously, c=1, so the result of division of 7 by 7 is 1.

Now let a=infinity and b=infinity. We have to find c such that infinity=infinity * c.

Obviously c=1. But, c=2 will also do since infinity*2=infinity. And c=3, c=4, c=4392123249543 will also do. Even c=infinity will be OK since infinity*infinity=infinity. So far we have that infinity / infinity = 1 and infinity / infinity = infinity, and by combining this two equalities we conclude that 1=infinity. And that obviously isn't true.

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Aug 22 2011: It says on your profile that you are an English Teacher. Why this interest in maths and infinity?
Another great example i forgot to give last time is
"1/(x^n)" where "x" and "n" are two positive integers. Keeping the value of "x" fixed, go on increasing the value of "n". the value of "1/(x^n)" becomes zero only if the value of n becomes infinity.

There are loads of such examples in mathematics.
• #### Sharon Turner

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Aug 23 2011: Thanks for the example. Yes I am an English teacher but in my free time I love reading about physics, mathematics, science, art. The maths and physics just fascinate me and the more I learn the more I want to learn. During my school years these subjects never made any sense to me because of the way they were taught. I can't just learn a formula I have to know why each letter is there. The history of the formula etc and then I can use it. So now I study it in this way and it makes sense. I just love these topics.)

• #### Sharon Turner

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Aug 23 2011: Thanks again Ed. I will definitely check these out. You seem to know a lot about sufism. Is this something you are interested in?
• #### Joe Fletcher

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Aug 21 2011: Its assuming you and the Earth lived forever. If you were to travel around the universe you would also do this. Its more to do the nature of curved space, and I believe in reference to the universe its due to gravitational warping.
I'm an anthropology major not a physicist, I just read on physics. You should cheak out A brief history of time (kinda outdated) the elegant universe, or many other qualatative based physics books for a MUCH deeper understanding of topics like this.
• #### Sharon Turner

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Aug 21 2011: Hi Joe,

Thanks for the reply and the book recommendations. I like you am also not a physicist and just read for pleasure. I have been reading a few other books you might be interested in:

-Parallel Worlds: The Science of Alternative Universes and Our Future in the Cosmos by Michio Kaku
-The Lightness of Being by Frank Wilczek
-Einstein's Unfinished Symphony: Listening to the Sounds of Space-Time (Joseph Henry Book) by Marcia Bartusiak and National Academy of Sciences

I don't know if you wold be interested in sharing insights/ideas or questions as you read various books.
• #### Joe Fletcher

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Aug 21 2011: Ok sounds good thanks for the recommendations

• #### Sharon Turner

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Aug 21 2011: Thanks Ed for your inside out play on this question. Very nice. I have a question though what kind of existence would you say was INFINITE? .)

• #### Sharon Turner

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Aug 21 2011: Hi Ed,

I also share an allergy to the word God and also religion. As for the Sufi, not yet. However one of my summer reads has been the RED BOOK by Rumi and I would also highly recommend Elif Shafak's book: The Forty Rules of Love based on the 40 Sufi tenants. I also loved this quote by Dr D. R. Dawkins when reading his latest. However I still feel the use of 'Supreme' in this quote has tones of religion even though he is not being religious. Reading your comments about the limits of English got me thinking about logographic scripts such as Chinese and how they view infinity beyond letters. Perhaps silence instead of words might be a better way to describe the infinite.
• #### Joe Fletcher

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Aug 21 2011: You can have an infinite border of a curved finite space, such as the surface of the Earth which can be traveled around infinitely. Applys to any curved border, thats what wedding rings symbolize ;–)
• #### Sharon Turner

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Aug 21 2011: I really enjoyed this answer Joe, the infinity of curved finite spaces and the reference to wedding rings. Although it also led me to ponder whether it can be travelled around infinitely. The Earth will not always be thereso does this make the border infinite due to the very nature of a finite space?! .)
• #### Carlin Covey

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Aug 21 2011: Sharon,

I think the simplest answer is "Yes, the mathematical concept of infinity does exist." It exists as a useful mathematical construct, just as imaginary and numbers exist as useful mathematical constructs. But you are never going to purchase an infinite number of apples (or an imaginary number of apples), so outside of mathematics these constructs are not as useful.
• #### Sharon Turner

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Aug 21 2011: Thank you Carlin for this definition. Your answer reminds me of Tosun Terzioğlu's (professor of mathematics at Sabanci University) comment that he wouldn't like to imagine what mathematics would be like if the Reimann Hypothesis was wrong.
• #### Carlin Covey

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Aug 22 2011: Well if you understand the Reimann Hypothesis, you obviously don't need my amateur definition of infinity. When you asked whether infinity exists, did you have a more metaphysical meaning in mind?
• #### Mountain River

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Aug 20 2011: Although supposedly there is minimum distance a particle can travel, space can be divided indefinitely.

If you have two singularities that are next to each other, you can never make them touch no matter how close you push them together because they have no size... I know that doesn't make any sense. If you think it does it shouldn't because it's a paradox.

You take it for granted that you can touch your nose, but a room full of the smartest men who ever lived would not be able to tell you exactly how you did it... And they-have-tried.

In a nutshell, if you had a string of space you could keep cutting it in half... forever!
• #### Sharon Turner

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Aug 21 2011: Thank you Borrah for this. You comment about cutting a string of space forever reminded me of a book I am reading:Parallel Worlds: The Science of Alternative Universes and Our Future in the Cosmos by Michio Kaku. He talks about expansion and the creation of other universes ( I highly recommend chapter 4). As for touching my nose I am going to go and read about that.)

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Aug 20 2011: Basically Infinity is a large quantity. It is also something that we cannot define properly. the best example of infinity is
"tan (90°)" check it out in Wikipedia. It is explained in the page http://en.wikipedia.org/wiki/Trigonometry.

Anything divided by zero [n/0] is also taken as infinity. But it actually should be "undefined". Think of it this way. when you have a fraction "3/4" you will say it is "3" parts out of "4". but when you have "n/0" it means "n" parts out of "nothing" which is undefined. But mathematicians like to take it as infinity
• #### Sharon Turner

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Aug 21 2011: Thank you Sahyadri and the tan (90) is an interesting example. 'undefined' is a nice way to explain the meaning of infinity. I have realised why reading and replying to comments how much we take the word 'infinity! and its derivations at face value. I wonder how infinity is described in logo graphic scripts such as Chinese.
• #### Goran Malic

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Aug 26 2011: It is not true that infinity cannot be defined properly. It is defined as the cardinality or size of the set of all integers. :) To be more precise, countable infinity is defined as the cardinality of the set of all integers. Uncountable infinity is defined as the cardinality of the set of real numbers.

As for the division by zero example, I wouldn't say that mathematicians like to take it as infinity. They take it as infinity in terms of limits, i.e. when it is possible, "n/0" is replaced with "the limit of n/x as x approaches zero" which indeed is infinity.