Why can't we define sequential numbers on the real number line?

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"Infinitely-many zeroes" means there are infinitely-many zeroes; the listing of zeroes never ends, so it cannot "end, followed by a 1". Since there are infinitely-many zeroes, there is no "end" to the zeroes, because the zeroes continue into infinity, where there is no end.

As soon as you tack on "...001", you have indicated a finite end to the zeroes, which is followed by a terminal 1. As soon as you put this end to the zeroes, you have a finite-digit number. There is no more "infinitely-many" any more.

I think the issue boils down to not understanding what is meant by "infinitely-many", and trying to equate 4.001, 4.0001, 4.000[some really big number of zeros]01 with an infinite decimal expansion. But the finite expansion 4.00[natural-number count of zeros]01 is not the same as the infinite expansion 4.000... or 4.000[lots of zeroes]0100....

Eliz.
 
stapel said:
"Infinitely-many zeroes" means there are infinitely-many zeroes; the listing of zeroes never ends, so it cannot "end, followed by a 1". Since there are infinitely-many zeroes, there is no "end" to the zeroes, because the zeroes continue into infinity, where there is no end.

As soon as you tack on "...001", you have indicated a finite end to the zeroes, which is followed by a terminal 1. As soon as you put this end to the zeroes, you have a finite-digit number. There is no more "infinitely-many" any more.

I think the issue boils down to not understanding what is meant by "infinitely-many", and trying to equate 4.001, 4.0001, 4.000[some really big number of zeros]01 with an infinite decimal expansion. But the finite expansion 4.00[natural-number count of zeros]01 is not the same as the infinite expansion 4.000... or 4.000[lots of zeroes]0100....

Eliz.
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I am certainly not trying to equate a very large number with "infinitley many". Clearly a very large number is not infinite.
 
Mates said:
Yes, I realize now that I was trying to redefine the reals to something that already exists, namely the hyperreals as I have been told.
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Some of us find calculus to be more intuitive using the hyperreal number system than using the real number system. As zermelo indicated that is called non-standard analysis.
 
Otis said:
That's a very good viewpoint. :)
[imath]\;[/imath]
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I don't like that argument because it doesn't not hold up for various reasons. In a Reiman sum, an infinite number of subintervals can end.

Similarly, the real number line can have an infinite number of numbers in an interval.
 
Mates said:
In a Reiman sum, an infinite number of subintervals can end.

Similarly, the real number line can have an infinite number of numbers in an interval.
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Kindly please provide an example of an infinite sum which ends after a certain number of summands.

Mates said:
Similarly, the real number line can have an infinite number of numbers in an interval.
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Yes, a finite-length interval on the real number line does indeed contain infinitely-many numbers within it. No, this is *not* the same thing as trying to claim that a finite-digit decimal expansion somehow also contains infinitely-many digits.

Eliz.
 
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stapel said:
Kindly please provide an example of an infinite sum which ends after a certain number of summands.
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I said that an infinite number of subintervals can end, not "a certain number".

Yes, a finite-length interval on the real number line does indeed contain infinitely-many numbers within it. No, this is *not* the same thing as trying to claim that a finite-digit decimal expansion somehow also contains infinitely-many digits.
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I am not claiming that a finite-digit decimal expansion contains infinitely-many digits. That would be self contradicting.
 
Mates said:
I don't like that argument because it doesn't [hold up] for various reasons.
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Please excuse me, Mates; I'd made the comment with the set of Real numbers in mind. I ought to have specified that it doesn't apply to the objects that you have in mind. Sorry 'bout that.
[imath]\;[/imath]
 
Otis said:
Please excuse me, Mates; I'd made the comment with the set of Real numbers in mind. I ought to have specified that it doesn't apply to the objects that you have in mind. Sorry 'bout that.
[imath]\;[/imath]
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I suppose the question becomes: can we "zoom in on" the reals like we can with the naturals?
 
Mates said:
I suppose the question becomes: can we "zoom in on" the reals like we can with the naturals?
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I am not sure what you mean by that question, but I would answer that by saying that we can "zoom in" on real and rational numbers in a way that we cannot for natural numbers. There is no such thing as a closest neighbor to a rational or real number. With the natural numbers, you cannot get more granular with respect to those numbers than with the successor relationship.

I apologize if I have misconstrued your question.
 
JeffM said:
I am not sure what you mean by that question, but I would answer that by saying that we can "zoom in" on real and rational numbers in a way that we cannot for natural numbers. There is no such thing as a closest neighbor to a rational or real number. With the natural numbers, you cannot get more granular with respect to those numbers than with the successor relationship.

I apologize if I have misconstrued your question.
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Sorry, I meant that we can "zoom in on" the Reimann sums to see the naturals in their successor relationship.

We can presumably assign a natural number to the very beginning of a Reimann sum. Then we can zoom in to see the next natural number. So the question (that has pretty much been answered) is can we zoom in on the reals the same way to see a next number.
 
Mates said:
Sorry, I meant that we can "zoom in on" the Reimann sums to see the naturals in their successor relationship.

We can presumably assign a natural number to the very beginning of a Reimann sum. Then we can zoom in to see the next natural number. So the question (that has pretty much been answered) is can we zoom in on the reals the same way to see a next number.
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Zooming in eternally on the real number continuum will show an unbroken line. This is the very nature of a continuum. As stated before, no matter how close we set two reals, there is always an uncountably infinite numbers between them.
 
MarkFL said:
Zooming in eternally on the real number continuum will show an unbroken line. This is the very nature of a continuum. As stated before, no matter how close we set two reals, there is always an uncountably infinite numbers between them.
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Yes, I would love to leave it at that, except when I think about a Reimann sum. The Reiman sum seems to have a very similar property in that no matter what 2 points we choose on some integral, we can also divide the distance in half to get another point.
 
Mates said:
Yes, I would love to leave it at that, except when I think about a Reimann sum. The Reiman sum seems to have a very similar property in that no matter what 2 points we choose on some integral, we can also divide the distance in half to get another point.
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With such a sum, we are taking a limit. In the same way a definite integral works.
 
Mates said:
Yes, I would love to leave it at that, except when I think about a Reimann sum. The Reiman sum seems to have a very similar property in that no matter what 2 points we choose on some integral, we can also divide the distance in half to get another point.
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I wonder if you are misunderstanding what a Riemann sum is:

en.wikipedia.org

Riemann sum - Wikipedia

en.wikipedia.org en.wikipedia.org

It is a particular finite sum of products representing rectangles. If you zoom in on the picture, you will eventually see just a part of one of the rectangles (or the edge between two).

You appear to think that a Riemann sum has infinitely many terms; but the definition of a Riemann integral is a limit, intended to avoid ever having to think about an actual infinity. Each Riemann sum is finite, and is just an approximation of the integral.

Yes, you can always get a new Riemann sum by subdividing the intervals; so what? This has nothing to do with your question.
 
Dr.Peterson said:
I wonder if you are misunderstanding what a Riemann sum is:
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Yes, I was definitely thinking of the Reimann integral, thanks.

Yes, you can always get a new Riemann sum by subdividing the intervals; so what? This has nothing to do with your question.
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Well isn't that the argument for not being able to have a next real number?
 
Mates said:
Well isn't that the argument for not being able to have a next real number?
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Yes, but it has nothing to do with Riemann sums (or integrals). It's just about how real numbers work.

I take it you accept that there is no next real number; what more is there to say about that?
 
Dr.Peterson said:
Yes, but it has nothing to do with Riemann sums (or integrals). It's just about how real numbers work.

I take it you accept that there is no next real number; what more is there to say about that?
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To be honest, I actually do not see how it is logical for there to be no next real number.

If we can think about the real number line geometrically, say, from 1 to 2, we know that there are only real numbers, no spaces. These numbers increase making them unique.

Now, if everything that I said is correct, then we can ask what is the closest thing to the number 1 in the interval? The answer should be another point/real since that is all there is. This point is larger than 1.

It's not a knock-out proof, but it is at least deductive reasoning.
 
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