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August 23rd, 2014, 11:13 AM   #11
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Sorry. Typo.
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August 23rd, 2014, 11:25 AM   #12
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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.
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August 24th, 2014, 07:00 PM   #13
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So, this question is not nonsensical.
The ray and line have length of the order aleph-sub-one, 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?
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August 24th, 2014, 10:25 PM   #14
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Originally Posted by shunya View Post
So neither the ray nor the line is bigger or smaller in length, than the other.

Am I correct?
At the risk of repeating myself, neither has a (well defined) length so we can't compare their lengths.
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August 24th, 2014, 10:35 PM   #15
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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.
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August 25th, 2014, 01:45 AM   #16
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I'm not sure I agree that the line and the ray have the same length.

I agree that a one-to-one 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 end-point and line without, because this is generally understandable by the lay-person. 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 non-zero 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.
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August 25th, 2014, 07:46 AM   #17
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Quote:
Originally Posted by shunya View Post
So, this question is not nonsensical.
The ray and line have length of the order aleph-sub-one, 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?
Only assuming the continuum hypothesis.

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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