Thread: Infinity in Mathematics is it real ?

I'm afraid I'm having a hard time understanding what exactly what you're wanting to discuss, or maybe what you're trying to say about your inability to grasp the concept. If infinity is a hard concept for you to truly grasp, I have to ask, is zero a concept you truly grasp? Can you really fathom what nothing is? You can observe the lack of something through the duality of existence... or a better way of putting it, you can know what it is like for something to not be there, because you can compare "empty space" to space that is occupied with something. This isn't what nothing is though. Nothing isn't something that exists as a comparison to something (I know I know, it doesn't exist at all, which is actually my point, lol), the way it does in our mind. In our mind, nothing is something that isn't there, not nothing. I feel like infinity is a far easier concept to grasp than nothing is. as a matter of fact, i don't really believe it is possible to understand what nothing is, because it neither is nor isn't. it's not even something we can begin to discuss. infinity is something that can be discussed endlessly, ironically enough.

Look at it this way. Infinity is exemplified very easily by making simple observations. When you measure the length of a line that is drawn to be 300cm long, you can measure 300 separate cm in that line. within those cm, you can find 10mm, making 3000mm in that line. you can keep going down the scale or up the scale and find a number we can assign to its length. now, following units of measurement isn't a great example given that the numbers are scaled linearly and it doesn't invoke a sense of infinity, rather readjusting one's perspective. however, you can stick to any scale you want. try finding an end to the number of decimal places you can reach trying to go between 1cm and 2cm. you can go on forever. you can always approach 1cm or 2cm, but you will never be able to reach a point where you can't go out another decimal place. but, as we all know, you can actually reach 1cm and 2cm. you can even reach 3, 4, 5, 10, 50, 1000, 100000, 1000000000000, etc. anyway you look at numbers and you try to escape infinity, you are met with the crushing reality that it cannot be escaped. there is no such thing as "finite" in the sense we use it. it is an arbitrary designation on any scheme above what we practically use it for. any time you supposedly have a finite range of numbers, you still have the infinite amount of decimals you can go in between them. it's no more finite than using whole numbers. in fact, they are the same as whole numbers depending on your perspective (or in this case, our scale/unit of measurement). what a decimal is for one set of numbers using a designated unit of measurement, is a whole number for another unit of measurement. same goes for using a different number base.

to elaborate on number bases a bit, we use a 10 based number system in our daily lives for mathematics. before reaching a point where we use "0" as a place holder for reaching the number of whole numbers that fill up one uh... "base", we have 10 numbers. binary that we use for computers is a 2 based number system, and in it there are no decimals. you can write every decimal using a 2 based number. as a matter of fact, you can write any number using any number base system. here's a good explanation on number bases

http://www.basic-mathematics.com/base-five.html

I think that is an interesting question. Infinity gets talked about a lot and we can't really see it in the "real world". Yet we can use the idea of infinity in physics to describe certain kinds of things we see in the world of physics.

Mathematicians study infinities and find out thing about it, so it really isn't "unknown" in that sense. Here's a video that explains some things about infinity and I think clears up some misconceptions.



You wanted to talk about infinity in physics and I found a small video that can give some very basic information on how it is used.



Sorry my response is mostly videos. I think they both do a good job at giving out the information and I think they will at least point you in the right direction.

I'm watching the first video now, and I can say this dude almost creeps me out... if I were able to feel creeped out. Maybe that's not the best way to describe it, but he looks like he's prone to... uh... cracking? Going crazy? Doing something wild and unexpected? It's funny, I've never really gotten this feeling from more than a handful of people before, lol. It's definitely his smile and how bugged out his eyes are.

Anyway, now that I've finished the video, I'm not necessarily sure how I feel about the information presented. A lot of it mirrored what I already posted, but some of it I need to look up because the video left me asking questions that were never answered. Unless my questions are already answered (which I assume they indeed are, and the dissonance here is caused my lack of understanding... or rather lack of knowledge, leading to a lack of understanding) else where, I'm not sure I can buy into the theory that there are some infinities that are larger than others. I feel like the arguments presented and the examples given were more a result of the semantics our language used to describe the math clashes with the actual math.

For instance, what he presented as the "bigger infinity" is only came about as a conclusion due to the inadequacy of viewing any given number from a single perspective. That is to say, 0.5 can be written as 1/2, 0.5, in base 5 it is written as 0.2, etc. You can write a number however you need to write it, and you can assign units to any number so as fix certain "impossibility" errors you would get from his diagonal number he changes based on a rule that wouldn't be listed, and it would allow for that number to be listed when previously it could not have been based on the limitations assigning language to mathematical concepts imposes. Using a single number system to describe or write a number also poses limitations, using a single base number system as and its inherent inability to list all fractions in a given range due to the inherent nature of infinity is flawed. Expanding the given range allows you to reach all real numbers. You can impose finite limitations on the number line, and you will never reach the beginning, nor the end no matter which end you start counting from. His logic is faulty two fold, as I see it. A) you could never reach any new number using his logic when trying to meet either end of a finite range. How can you go from 0.1 to 0.2 when there are an infinite number of decimals between them? 0.100000000000000001, 0.0001, 0.111111111111111, 0.1212343, etc. There is never a point you could change numbers at all based on that type of thinking. The reason I find for this "paradox" is one that actually explains that it is no paradox at all. The premises behind the argument are incorrect, which is what leads to this mind boggling issue of seemingly being unable to ever reach a different "number".

The reason here is because infinity is actually seen as a preventing factor in ever reaching the beginning or end of a finite range. By the very nature and definition of the concept, the beginning and ending numbers must be included in the range you have chosen. Just because you could endlessly be writing out decimals between 1 and 2 doesn't mean that "infinity" does not include 1 and 2. To suggest otherwise means a fundamental misunderstanding of what infinity truly "is" must be occurring somewhere. The other idea is that any a given range is actually finite in itself. Functionally it makes sense to call writing all the possible numbers 1 and 2 "finite", but really it isn't. What are 1 and 2 over 10 and 20, 0.1 and 0.2, or any different set of numbers you can apply a given rule to that you apply to all numbers in between? You have an infinite set of choices before you in making different numbers to chose between. Say you square all numbers in your range, you wind up with the decimals and numbers between 1 and 4. You might say that introducing the rule means introducing new numbers to the list, but truth be told I don't see why the results of any possible rules applied to the numbers in a given range can't be included part of that range. When you run the rule backwards you can return to what you would call as the "smaller infinity" that was originally generated by the numbers and decimals between the original range. Their values are all equal, yet given the possibility of an infinite number of rules for changing the numbers, every number you could possibly encounter can be found in that list. Trying to call one infinity larger than another infinity because what our intuition says about the different examples we can generate that seemingly allow for a greater size of infinity to exist is flawed. That conclusion can only come about if you willingly leave out numbers from the list that should be included in the list in the first place.

I'm foreseeing arguments that my argument basically states that no given numerical value can be greater or less than another, but that simply isn't true. 10 is still greater than 5, 1 still less than 3. I mean, what's the difference between saying that and trying to list all the numbers between 1 and 10, and 1 and 5? Shouldn't there be more between 1 and 10 than between 1 and 5? While it's true that there are more numbers between them, if say, we count by 1. You have 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 in the range of 1 and 10, and 1, 2, 3, 4, and 5 between 1 and 5, and we can easily count that there are 5 less numbers between the two ranges. However, my argument doesn't make this observation false. If you noticed what happened when I listed those whole numbers, I was purposefully excluding numbers that should be included in these lists (when trying to infinitely list numbers in between) because they did not fit my "rules" for being written. I am not dealing with infinity, I am dealing with whole numbers. When it comes to whole numbers, there are twice as many between 1-10 than 1-5. The guy in the video said it himself. Infinity is not a number, it is a concept. In order for the defining criteria of infinity to be met, or in other words for him to use the word infinity accurately in his examples, he must include every number possible in his range, no matter your start point or end point. The starting point and ending point do not affect the numbers in a list when you are truly making an infinite list. Believing some numbers are missing is merely an oversight caused by not thinking of what you are truly doing all the way through. When it comes to infinity, because it is not a finite value represented by a symbol of our choosing (it is a symbol representing an infinite value, after all) it is not bound to the rules and logic that follows for symbols representing finite values (like 1, 5, 0.67, 2/3, 7 to the 8th power, etc.). Any finite starting and ending points in a range where we list all the infinite numbers contained between are as good as illusions. No matter how you slice looking at infinity, it's always going to be infinite in every aspect you look at, at any perspective you view it from. It will have infinite decimals (divisions) in between, infinite powers, infinite multiples, infinite differences, infinite additions, infinite anything you can think of. You can make any combination of those functional operations that you want, there are and infinite number of them, and there are an infinite number of answers for those infinite operations.
The second you talk about counting infinitely between something, you are talking about all numbers possible. Or more precisely, a value that never ends (and even never begins in a way, lol). No matter what, you will find the same numbers in one of these supposed "smaller" infinities that you find in a larger infinity.

edit: a few of my arguments are poorly written and some are invalid, after rereading what I wrote. I'll try to compile a more concise, and valid argument here in a while.

Hmm, I'll try to address some of the points you made, I probably won't get all of them, and I also have a bit of a limited understanding of this topic myself, but I'll do my best. Here we go.

First I want to talk about the point about expressing numbers in different ways. Some decimals do not repeat at all, and cannot be expressed as fractions. These numbers are called irrational numbers. The point I think the video was making is that there will always be another irrational number you can add to a list of decimal numbers. Even changing the base of the numbering system will not work. Pi in binary and any other base does not repeat or have any pattern we know of.

Regarding how there are an infinite amount of decimals between 0 and 1 and thus could never reach 1 from zero, that is one of the points the video address I think. In this field of mathematics the listing of numbers is called "cardinality". So the set "1,2,3,4,5..." and the the set "2,4,6,8,10..." are the "same size" of infinity, so it's said that they have the same cardinality, they can be put into a 1-1 relationship with each other. What the video was saying is that all the decimals between 0 and 1 can't be listed, so that set doesn't have the same cardinality. The reason it's called "bigger" is because the proof used shows that you can always create one more irrational number, a new element, from the counted list of decimals. You could do it forever, always using new irrational products to create even more unlisted irrational numbers. It show they are not the same kind of infinity. The term larger is used more in the colloquial sense, but I think you can see why it is used. If you want to read more about cardinality here is a link to Wikipedia that explains the bare bones of it. It's sort of math heavy though.

When it comes to having to use all the numbers possible in a range to be truly infinite, I don't think that is the case. What we are talking about are different sets of numbers. Each set has a rule, something that defines the relationship between one element and the next. In the natural numbers "1,2,3..." that is n+1. For even numbers it's n+2, and so on. Each of these sets can go on forever. There would never be a last term, a new one can always be made. If we have a set that has the rule 1/n, it will approach a number, in this case 2, when we go to infinity, but there will still be an infinite amount of terms, you can always add one more, no matter how large or small. That is what the concept of infinity is, an ability to always add one more element. What I think you are talking about is about how to divide the number line. There are many ways to do it, and we can choose ways that have no decimals and that would still go on to infinity.

When it comes to the other points, I didn't really understand your arguments. If you have the energy, could you try to restate them? I'll do my best to answer you. And believe you me, I'm no mathematician. I only have a pretty basic idea of these things myself. I know relying on authority is a weak argument, but Cantor's uncountability proof is seen as pretty solid and the people saying this are the kind who do math for a living. I hope we can talk more about this and get a better grasp of it in the process.