August 23rd, 2014, 11:13 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,638 Thanks: 2623 Math Focus: Mainly analysis and algebra 
Sorry. Typo.

August 23rd, 2014, 11:25 AM  #12 
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 390 Thanks: 70 
No problem, Archie. I would like to mention that I just can't get my mind around the fact that there is an infinite number of rationals ($\aleph_0$) and that there is a BIGGER infinite number of reals ($\aleph_1$). Psychologically, I would like to deny this, but I have studied the proof, and it is valid. 
August 24th, 2014, 07:00 PM  #13 
Senior Member Joined: Oct 2013 From: Far far away Posts: 422 Thanks: 18 
So, this question is not nonsensical. The ray and line have length of the order alephsubone, using the analogy of comparing all the reals with the positive reals (including zero). So neither the ray nor the line is bigger or smaller in length, than the other. Am I correct? 
August 24th, 2014, 10:25 PM  #14 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,638 Thanks: 2623 Math Focus: Mainly analysis and algebra  
August 24th, 2014, 10:35 PM  #15 
Senior Member Joined: Jul 2014 From: भारत Posts: 1,178 Thanks: 230 
Since both are infinite, both go endlessly and one cannot say which is bigger, but still, if both go till infinity, the rate of increase of both should be specified. If both increase by the same rate then I would say that a line is bigger.

August 25th, 2014, 01:45 AM  #16 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,150 Thanks: 730 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
I'm not sure I agree that the line and the ray have the same length. I agree that a onetoone correspondence can be applied between positive integers and positive even integers, so they must have the same number of elements, but the length of something isn't defined by the number of elements defining the course of the line, it is the sum of all intervals between adjacent elements over the total span of the object. Since the distance between adjacent integers is 1 and the distance between adjacent even integers is 2, I would argue that the latter is twice the length of the former. Just to be clear, my previous post used the idea of 'direction' to convey the differences in length between a line with an endpoint and line without, because this is generally understandable by the layperson. I guess a more specific (and clearer) way of conveying that is to say that if the ray is defined to be the set of positive integers (including 0), the line is defined to be the set of all integers and n is the number of nonzero positive integers, then: Number of elements of ray: n + 1 Number of intervals of ray: n Length of one interval: 1 Length of ray: n Number of elements of line: 2n + 1 Number of intervals of ray: 2n Length of one interval: 1 Length of line: 2n As $\displaystyle n\rightarrow \infty$, the lengths of both lines tend to infinity, but that doesn't magically make both lengths the same. I agree with the previous post stating that it doesn't really make sense to discuss the length of infinite objects, but the purpose of this exercise anyway is to demonstrate that $\displaystyle \infty$ is not really a number and must be considered carefully in every problem involving it. Last edited by Benit13; August 25th, 2014 at 01:47 AM. 
August 25th, 2014, 07:46 AM  #17  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
There is a bijection between the points on a ray and the points on a line, so the two have the same cardinality. But this is only $\aleph_1$ under CH. It's unconditionally $2^{\aleph_0}=\beth_1.$ But it's not clear if the original question is asking this or something else.  

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