October 16th, 2014, 09:17 PM  #11 
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  The process is no different if the infinity symbol is used or not. The point is that you don't really use infinity. You find the limit by examination of the expression as x increases. Using the word infinity is misleading as it is not what is really happening.

October 16th, 2014, 09:22 PM  #12 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,935 Thanks: 1129 Math Focus: Elementary mathematics and beyond 
That is what's happening! The statement of the limit says "we can make f(x) as close to 1 as we like by choosing x sufficiently large". There is no restriction on the size of x. If there were, we'd have a constant, not a limit. Last edited by greg1313; October 16th, 2014 at 09:40 PM. 
October 16th, 2014, 09:39 PM  #13  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  It gives a solution for natural numbers up to n. It does give a solution for the completed set of all natural numbers. In this respect it is not a complete solution if infinity is admitted. Quote:
 
October 16th, 2014, 09:48 PM  #14 
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  
October 16th, 2014, 10:02 PM  #15  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
The problem is at a much more basic level. If something is endless or limitless then by definition it cannot have an end point. The concept of a completed infinity implies an end. If something is truly endless then it can never be completed. And so saying something has no limit does not mean it is infinite. Furthermore, consider the expression 'as x approaches infinity'. Assuming we can use infinity in this context, then it does not matter how much we increase x because we will still be an infinite distance away from infinity. The same applies if we decrease x or leave x unchanged. Since increasing or decreasing or not changing x all result in being the same distance from infinity, it follows that we cannot 'approach' it. Last edited by Karma Peny; October 16th, 2014 at 10:43 PM.  
October 16th, 2014, 10:40 PM  #16 
Senior Member Joined: Aug 2012 Posts: 2,311 Thanks: 706 
One can employ infinity for its usefulness, even while denying it any ontological status. That is, one can if one likes adopt the following position: "I know that infinite numbers do not exist; and if they did exist I'd stamp them out; for I dislike them. Yet, they are curiously useful and make many computations simpler in physics, mathematics, biology, and economics. So I will freely use them, even though they do not exist." In fact that's a familiar sounding position. Some people felt that way about the idea that the earth revolves around the sun. Of course it doesn't, everybody knows that God put the earth at the center of the universe. But the model  false though it is  with the sun at the center of the solar system sure does simplify the calculations. So let's just use that system in astronomy class, even when we know it's not literally true. And that "imaginary" number i, with the property that if you square it you get 1. That doesn't exist either. Of course it does allow us to solve polynomial equations, and it makes the calculations of electromagnetic theory far simpler ... so let's just take it as a useful fiction. OP, would that satisfy you? From a philosophical point of view, what does "true" mean, anyway? Isn't truth just what we all find convenient to believe? You could in fact work out all the math with the earth as the center of the universe. The universe wouldn't change, only the math would. I don't care if the engineers building the bridge I'm driving on believe in infinity. I just expect that they've studied calculus. 
October 16th, 2014, 11:20 PM  #17  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Quote:
I think you are struggling with exactly the same problems with infinity as me. I looked at your articles, and I think the problem lies in what you have written under the topic "Removing infinity from repeating decimals", you write that: 0.999… = 1 this is incorrect (yet many mathematicians accept it) lim(0.999…) = 1 this is correct You also write that " we can say that the limit of 0.333… as the number of decimal places increases, equates to one third. This is completely different to saying that 0.333… equals one third, which would be wrong, as it never does." If you look at my last thread on this forum "What lies beyond infinity", I presented similar ideas there, but they were not accepted as valid. I did not suggest abandoning infinity. Instead, I was even more radical, I suggested abandoning the decimal number system altogether if we can't accept infinitesimals. In other words, I can deny that $\displaystyle $$\frac{1}{3}$ has a decimal number representation if the infinitesimals don't exist. $\displaystyle $$\frac{1}{3} = 0.333.....$ only if the infinitesimals exist, infinitesimals which are $\displaystyle \neq 0$. Otherwise, we are forced to admit that $\displaystyle $$\frac{1}{3} \approx 0.333.....$ Somehow I get the feeling that abandoning the decimal number system corresponds to your idea of abandoning infinity. We should abandon the infinite amount of decimals of $\displaystyle $$\frac{1}{3}$ if we cannot count them correctly. I see no other way out of this problem. Either we can count them all or we can't count them all.  
October 16th, 2014, 11:40 PM  #18  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
There are many processes that appear to use infinity, such as calculus, but the word is often used because people think of infinity as being an unimaginably large number. Calculus was not devised using infinity, the proof of the fundamental theorem does not require infinity, the use of differential and integral calculus does not require infinity. But still, such processes are claimed to justify the value of infinity. Gauss’s views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us to talk about limits. The notion of a completed infinity doesn’t belong in mathematics'. Quote:
In order to solve previously unsolvable quadratic equations, we assume that the square root of 1 can exist. This appears wrong as it breaks the rule that something times itself must be positive (but maybe this rule was wrong all the time). It soon becomes apparent that we can form a logically consistent set of mathematical rules that include the square root of 1. In a similar fashion, it is perfectly acceptable to assume a completed collection of an endless sequence can exist. But in this case it is not possible to form a logically consistent set of rules. Many paradoxes arise and it appears to be possible to prove that such an object cannot exist. Last edited by Karma Peny; October 16th, 2014 at 11:53 PM.  
October 17th, 2014, 12:16 AM  #19  
Member Joined: Oct 2014 From: UK Posts: 62 Thanks: 2  Quote:
I've just looked through that thread you mentioned, it's a long one! The real number line supposedly stretches from infinity to +infinity and any section of it, however small, contains an infinite number of numbers. As you might have guessed, I reject the whole basis of a continuum as it is based on the concept of infinity, which is nonsensical. I prefer to address the problem that the continuum is trying to provide a solution for. The problem, as I see it, is how can we fully express irrational numbers in a framework that we can work with? The solution is to use symbols, such as π and √2 rather than their decimal expansions. We can then work with irrational numbers in an abstract framework. If we want to see a numeric result rather than a set of expressions containing symbols then we have to expand the expressions using the constraints of the real or abstract world to which we are applying it. This problem has already been solved to some extent for the encoding of vector graphics in computer software. Vector images are made up from multiple objects. Each object consists of mathematical instructions that define shapes. And each shape is defined in terms of points and paths. This makes the image fully scalable without loss of quality. It only gets converted to discrete pixels when the image is rendered onto a real world object like a small screen or the large side of a building. Last edited by Karma Peny; October 17th, 2014 at 12:21 AM.  
October 17th, 2014, 01:34 AM  #20 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,159 Thanks: 878 Math Focus: Wibbly wobbly timeywimey stuff.  

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