Concerns about the geometry of the real number line

[StringJunky]

4 hours ago, Genady said:

Yes, agree. Another +1 to @wtf.

The sad part is that here are 100 posts, and 0 progress.

Sometimes things take a little longer...

[Genady]

11 minutes ago, StringJunky said:

Sometimes things take a little longer...

I know, I know. That's why I could never be a teacher.

[Boltzmannbrain]

On 3/11/2023 at 5:54 PM, wtf said:

Because the transfinite ordinals and cardinals are not real numbers. The subject of the thread is "The geometry of the real number line." That's YOUR topic, right? So we are discussing the real numbers. The transfinite ordinals and cardinals are fascinating in their own right, but have nothing to do with the real numbers. In fact the transfinite ordinals (and the cardinals, which are technically a proper subclass of the ordinals) do not intersect the real numbers at all. The transfinite ordinals and cardinals are neither subclasses nor superclasses of the real numbers.

They're just a completely different subject. If you are trying to understand whether there's a largest number in [0,1), it's no help to think about transfinite numbers, since transfinite numbers are not in that interval at all. 

Does that make sense? 

Besides, haven't you already said you are only interested in the standard real numbers? Why are you suddenly interested in mathematical objects that are NOT standard real numbers?
 

 

The reason why I am backtracking a bit on this issue is because I am not using an infinite number in place of a real number.  

 

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This para does not make sense. First, we can't use transfinite numbers in a discussion of the reals, because transfinite numbers are not members of the real numbers. We can of course use transfinite numbers to talk about the cardinality of various subsets of the reals, but that's not what we're talking about here.

 

 

 

 

But that is how I am using an infinite number.  Since there are an infinite number of numbers/points or decreasing subsets from 0 to 1, then it would seem fair to use an infinite number where it is relevant.

 

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In your imagination. But the proof that there is no largest number in [0,1) should cause you to realize that your intuition is flawed. It should give you a better intuition.

Now there is nothing wrong with having such a faulty intuition. Pretty much everyone has faulty intuitions about the real numbers before they see these technical discussions. But now that you've seen a formal proof that there is no largest number in [0,1), you should be willing to realize that your intuition is faulty, pre-mathematical as it were, and you should update your intuition.

 

 

Yes, I fully agree.  Intuition is no argument here.

 

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What exactly about the proof are you still unconvinced about? I asked you that in my previous post. It's no good for you to say you're unconvinced, without saying exactly what aspect of the proof you are unsure about. 

 

I briefly mentioned it earlier, but I will say it again.  

The "halving" proof definitely works for each individual case, but what about applying it an infinite number of times?  But let's move on because I think the other proof you put below is not challengeable.  So regarding proofs for this, I will accept that they exist and are unchallengeable.

 

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We are not going to discover a largest number in [0,1), because we already proved a few posts back that there is no such thing. 

Now I'm confused. Didn't you see and more or less agree with the proof I already posted?

 

 

I did not start this thread to disprove anything.  I started this thread to say that it seems illogical for the reals to have no next number or no smallest number (smallest real is what all of this really boils down to).

The reductionist in me wants to define the reals to have a smallest number sort of like the naturals (and then maybe we wouldn't have to have the infinitesimal hyperreals).  But this is just a side thought. 

Edited March 13, 2023 by Boltzmannbrain

[Genady]

In spite of the title, I didn't see anything in this thread that pertains to real numbers. All the arguments, such as non-existence of the next or the smallest numbers, equally apply to rational numbers. Ancient Greeks could discuss and resolve it.

[wtf]

4 hours ago, Boltzmannbrain said:

But that is how I am using an infinite number.  Since there are an infinite number of numbers/points or decreasing subsets from 0 to 1, then it would seem fair to use an infinite number where it is relevant.

To strain an analogy, when you buy a dozen eggs, there are 12 eggs in the carton. But when you open up the carton you do not find the number 12. You find 12 eggs. 

Likewise there are uncountably many real numbers, \( 2^{\aleph_0} \) in fact, in the interval [0,1], or in any other interval of the reals. But you will not find \( 2^{\aleph_0} \) among the real numbers. Hope this is clear. 

 

 

 

4 hours ago, Boltzmannbrain said:

The "halving" proof definitely works for each individual case, but what about applying it an infinite number of times?

You've asked this several times, and each time you get the same answer. If you start from x = 1/2, repeating the halving process gives an infinite sequence of real numbers 1/2, 3/4, 7/89, 15/16/ 31/32, etc. None of them are the largest number in [0,1).

In fact this example provides a nice intuition for the fact that in [0,1), the point at 1 is a limit point of the set. There is an infinite sequence of points getting closer and closer to 1, but there is no last point in the sequence. 

If it helps visualization, think of it the other way. Start with 1 and keep halving: 1, 1/2, 1/4, 1/8, 1/16, ... There are infinitely many numbers in the sequence. There is no "last" number. This shows in fact that there is no smallest positive real number, just an infinite sequence of smaller and smaller positive reals that approach 0 as a limit.

 

 

4 hours ago, Boltzmannbrain said:

But let's move on because I think the other proof you put below is not challengeable.  So regarding proofs for this, I will accept that they exist and are unchallengeable.

Ok, good. Does any of this help your intuition? One purpose in studying the formal properties of the real numbers is to improve our intuitions about them. 

 

4 hours ago, Boltzmannbrain said:

I did not start this thread to disprove anything.  I started this thread to say that it seems illogical for the reals to have no next number or no smallest number (smallest real is what all of this really boils down to).

You mean it's counterintuitive, and that's normal. Many people have an intuition that there's a smallest positive real, until they are shown the proof that there isn't.

But in terms of logic, it's logic that shows that there can be no smallest positive real. That's because the real numbers are a field. In a field we can add, subtract, multiply, and divide (except by 0) any two numbers.

So if someone claims that, say, \( x \) is the smallest positive real number, it's immediately clear that this is false; because \( \frac{x}{2} \) is also a positive real and it's strictly smaller than \( x \). 

You can see that, can't you? In terms of logic, we know that there can not possibly be a smallest positive real. Another example is the sequence .1, .01, .001, .0001, etc. The sequence goes on without end. There is no smallest real number.

 

 

4 hours ago, Boltzmannbrain said:

The reductionist in me wants to define the reals to have a smallest number sort of like the naturals (and then maybe we wouldn't have to have the infinitesimal hyperreals). 

On the contrary. The hyperreals are also a field. They are an alternative model of the exact same first-order axioms that the reals are a model of. So if you claim that x is the smallest positive hyperreal, then x/2 is a smaller positive hyperreal.

There is no smallest number, not even an infinitesimal, in the hyperreals. You can always divide a hyperreal infinitesimal by 2.

Edited March 13, 2023 by wtf

[Genady]

Here is a shorter proof:

If q is the smallest positive number, then 1/q is the largest positive number. But there is no largest positive number because we can always add another positive number. Thus, there is no smallest positive number.

[studiot]

14 hours ago, Genady said:

Here is a shorter proof:

If q is the smallest positive number, then 1/q is the largest positive number. But there is no largest positive number because we can always add another positive number. Thus, there is no smallest positive number.

 

This notion has already been posted and ignored.

However there is more mileage in the reciprocal idea if we perform our analysis in terms of neighbourhoods.

We can, for instance introduce the "neighbourhood of infinity".

 

 

19 hours ago, Boltzmannbrain said:

I did not start this thread to disprove anything.  I started this thread to say that it seems illogical for the reals to have no next number or no smallest number (smallest real is what all of this really boils down to).

The reductionist in me wants to define the reals to have a smallest number sort of like the naturals (and then maybe we wouldn't have to have the infinitesimal hyperreals).  But this is just a side thought. 

 

19 hours ago, Boltzmannbrain said:

The reason why I am backtracking a bit on this issue is because I am not using an infinite number in place of a real number.  

 

Backtracking and side thoughts apart, there is much to be said for the concept of neighbourhoods, in particular the neighbourhood of infinity as noted above.

This helps avoid making ridiculously self contradictory phrases such as "infinite number".

Paragraph D from Thurston below takes pains to point out that infinity only behaves as a number in some ways

 

 

[Boltzmannbrain]

1 hour ago, wtf said:

To strain an analogy, when you buy a dozen eggs, there are 12 eggs in the carton. But when you open up the carton you do not find the number 12. You find 12 eggs. 

Likewise there are uncountably many real numbers, 2ℵ0 in fact, in the interval [0,1], or in any other interval of the reals. But you will not find 2ℵ0 among the real numbers. Hope this is clear. 

 

I am interested in this geometrically.  So I would like each number to represent a point/position in a space or on a line.

 

Something interesting that keeps coming to me is when thinking about the Reimann integral.  There are n partitions, and n eventually equals infinity in order for the sum to be complete.  In this case there is a 1st, 2nd, 3rd etc. point/number (with a column) that "fills" in a real space.  I will read more about how that is possible and how it can help me figure all of this out.   

 

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You've asked this several times, and each time you get the same answer. If you start from x = 1/2, repeating the halving process gives an infinite sequence of real numbers 1/2, 3/4, 7/89, 15/16/ 31/32, etc. None of them are the largest number in [0,1).

In fact this example provides a nice intuition for the fact that in [0,1), the point at 1 is a limit point of the set. There is an infinite sequence of points getting closer and closer to 1, but there is no last point in the sequence. 

If it helps visualization, think of it the other way. Start with 1 and keep halving: 1, 1/2, 1/4, 1/8, 1/16, ... There are infinitely many numbers in the sequence. There is no "last" number. This shows in fact that there is no smallest positive real number, just an infinite sequence of smaller and smaller positive reals that approach 0 as a limit.

 

 

I said to repeat it an infinite number of times, not a finite number of times.

 

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Ok, good. Does any of this help your intuition? One purpose in studying the formal properties of the real numbers is to improve our intuitions about them. 

 

My intuition has always been that you can't get to a largest number or a smallest number (using typical methods).  Now I am starting to rethink that intuition.

 

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You mean it's counterintuitive, and that's normal. Many people have an intuition that there's a smallest positive real, until they are shown the proof that there isn't.

 

I don't agree.  I don't think people believe in a smallest.

 

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But in terms of logic, it's logic that shows that there can be no smallest positive real. That's because the real numbers are a field. In a field we can add, subtract, multiply, and divide (except by 0) any two numbers.

So if someone claims that, say, x is the smallest positive real number, it's immediately clear that this is false; because x2 is also a positive real and it's strictly smaller than x. 

You can see that, can't you? In terms of logic, we know that there can not possibly be a smallest positive real. Another example is the sequence .1, .01, .001, .0001, etc. The sequence goes on without end. There is no smallest real number.

 

 

Yes it is definitely logical since I have yet to prove otherwise.

 

[wtf]

41 minutes ago, Boltzmannbrain said:

I am interested in this geometrically.  So I would like each number to represent a point/position in a space or on a line.

Yes, a real number represents a point, or the location or "address" of a point if you prefer, on a line. But in fact this is actually a circular definition, since in math "the real line" is just another name for the set of real numbers. There's not a separate thing called a line that happens to be indexed by real numbers. Rather, we construct the real numbers, and casually talk of "the real line" meaning exactly the set of real numbers. So there's a bit less here than meets the eye.

Now when you say space, I hope you do not mean physical space as in physics; because the mathematical real numbers are very different (as far as we know) from anything in the physical world. In physics we have the Planck length, which is a length below which our equations of physics are not applicable. That doesn't mean that physical space has a smallest distance. Rather, it just means that below the Planck length we can't reason sensibly about the nature of space. But either way, math \( \neq \) physics, and that's always good to keep in mind. We are talking only about mathematical space and nothing to do with the world we live in. Unless you're talking about physics, in which case that's a whole different kettle of fish, as they say.

 

 

41 minutes ago, Boltzmannbrain said:

I said to repeat it an infinite number of times, not a finite number of times.

 

1/2, 3/4, 7/8, 15/16 ...

Are there not infinitely many elements in that sequence? Isn't the halving process being carried out infinitely many times? I hope we can reach agreement on this point. There are infinitely many counting numbers 1, 2, 3, 4, ..., right? And for each counting number \( n \), we have an element of the sequence \(\frac{2^n - 1}{2^n}\). Right? Let's please nail this down. We are going "half the distance" an infinite number of times. Let me know if that's not clear because you seem to be disagreeing.

 

41 minutes ago, Boltzmannbrain said:

My intuition has always been that you can't get to a largest number or a smallest number (using typical methods).  Now I am starting to rethink that intuition.

 

Now I'm confused, because I thought you've been saying there is a largest number in [0,1), which amounts to saying that there is a smallest positive real number. If you don't believe this, and if you claim you have never believed this, then I wonder what we've been talking about. 

41 minutes ago, Boltzmannbrain said:

I don't agree.  I don't think people believe in a smallest.

 

You've been claiming there's a smallest positive real number in multiple posts. Am I completely misunderstanding you?

41 minutes ago, Boltzmannbrain said:

Something interesting that keeps coming to me is when thinking about the Reimann integral.  There are n partitions, and n eventually equals infinity

Sorry these two quotes are out of order but it seems to be ok.

No, the number of partitions is NEVER infinity. The number of partitions is always some finite natural number \( n \). As \( n \) gets large, the upper and lower Riemann sums may happen to approach the same number as a limit; and if they do, we call that number the (definite) Riemann integral. 

Maybe the wording's a little confusing. As we let \( n \) go to infinity, for each \( n \) we have a pair of numbers, the upper and lower Riemann sums; so we get two infinite sequences. If each sequence has a limit and the two limits are the same, we call that limit the Riemann integral.

But there are never "infinitely many" partitions. That's an informal way of talking about the limit. The entire idea of calculus is to use limiting processes to avoid talking about "infinitely many" partitions or "infinitely small" things.

 

 

41 minutes ago, Boltzmannbrain said:

in order for the sum to be complete.  In this case there is a 1st, 2nd, 3rd etc. point/number (with a column) that "fills" in a real space.  I will read more about how that is possible and how it can help me figure all of this out.   

We are not filling the space, we're approximating it with rectangles, and the limit of the approximation process is the area under the curve. It's a limiting process.

I'm not entirely sure what you mean that studying the Riemann integral will help you figure this out. You've been concerned with the question of finding a "last" number in [0,1), and you seem to agree that we have proved that there can be no such thing. Is there a different concern you have about something else? Integration can't help. If (in two dimensions, say) we include or don't include the boundary of a disk, the area is still the same. That's the point we made earlier, that the lengths of [0,1] and [0,1) are exactly the same, namely 1. Points have no length or area, so their presence or absence makes no difference to an integral.

Edited March 14, 2023 by wtf

[Boltzmannbrain]

19 hours ago, wtf said:

Yes, a real number represents a point, or the location or "address" of a point if you prefer, on a line. But in fact this is actually a circular definition, since in math "the real line" is just another name for the set of real numbers. There's not a separate thing called a line that happens to be indexed by real numbers. Rather, we construct the real numbers, and casually talk of "the real line" meaning exactly the set of real numbers. So there's a bit less here than meets the eye.

Now when you say space, I hope you do not mean physical space as in physics; because the mathematical real numbers are very different (as far as we know) from anything in the physical world. In physics we have the Planck length, which is a length below which our equations of physics are not applicable. That doesn't mean that physical space has a smallest distance. Rather, it just means that below the Planck length we can't reason sensibly about the nature of space. But either way, math ≠ physics, and that's always good to keep in mind. We are talking only about mathematical space and nothing to do with the world we live in. Unless you're talking about physics, in which case that's a whole different kettle of fish, as they say.

 

I meant a space in R².

 

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1/2, 3/4, 7/8, 15/16 ...

Are there not infinitely many elements in that sequence? Isn't the halving process being carried out infinitely many times? I hope we can reach agreement on this point. There are infinitely many counting numbers 1, 2, 3, 4, ..., right? And for each counting number n, we have an element of the sequence 2n−12n. Right? Let's please nail this down. We are going "half the distance" an infinite number of times. Let me know if that's not clear because you seem to be disagreeing.

 

 

If you half the distance an infinite number of times you get 0 don't you? 

 

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Now I'm confused, because I thought you've been saying there is a largest number in [0,1), which amounts to saying that there is a smallest positive real number. If you don't believe this, and if you claim you have never believed this, then I wonder what we've been talking about. 

You've been claiming there's a smallest positive real number in multiple posts. Am I completely misunderstanding you?

 

 

I only meant that my intuition has always been that there couldn't be a smallest number (as I believe everyone's intuition is), but now that has changed.

 

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No, the number of partitions is NEVER infinity. The number of partitions is always some finite natural number n.

 

Then what you said above cannot be true either about the halving process being carried out an infinite number of times, since n is always finite.

 

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As n gets large, the upper and lower Riemann sums may happen to approach the same number as a limit; and if they do, we call that number the (definite) Riemann integral. 

Maybe the wording's a little confusing. As we let n go to infinity, for each n we have a pair of numbers, the upper and lower Riemann sums; so we get two infinite sequences. If each sequence has a limit and the two limits are the same, we call that limit the Riemann integral.

 

 

Ok, I understand.  That is very interesting, thanks.  I never took the second semester of the advanced calculus course, so I never got a proper understanding of the more rigorous definition of the Riemann integral.  I only took the typical calculus for learning integrals.

 

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But there are never "infinitely many" partitions. That's an informal way of talking about the limit. The entire idea of calculus is to use limiting processes to avoid talking about "infinitely many" partitions or "infinitely small" things.

 

 

Then how can we get 0 when calculating the widths of the partitions over n as n goes to infinity?  What finite n value would allow this?

 

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I'm not entirely sure what you mean that studying the Riemann integral will help you figure this out. You've been concerned with the question of finding a "last" number in [0,1), and you seem to agree that we have proved that there can be no such thing. Is there a different concern you have about something else? Integration can't help. If (in two dimensions, say) we include or don't include the boundary of a disk, the area is still the same.

 

I am interested in the integral because of what we are discussing above.  It seems like that will help me understand the ideas of density, infinity vs limit to infinity, etc. from a different perspective, namely the integral (well to be more specific, the 1 dimensional aspect of the integral).

 

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That's the point we made earlier, that the lengths of [0,1] and [0,1) are exactly the same, namely 1. 

 

I would think so too.  Except when I think about that example I gave earlier about the segment not fitting because of one extra point, I am left confused and very curious.

Edited March 15, 2023 by Boltzmannbrain

[studiot]

21 minutes ago, Boltzmannbrain said:

It seems like that will help me understand the ideas of density, infinity vs limit to infinity, etc. from a different perspective,

You clearly don't want to understand since I offered you an alternative treatment using neighbourhoods and you ignored it totally.

 

The beauty of neighbourhoods is that it does not depend upon whether or not there is a smallest real number or a nearest real number or what happens at infinity.

And yet neigherbourhood analysis arrives at the same results as the other approaches you have been arguing about.

 

[Boltzmannbrain]

4 minutes ago, studiot said:

You clearly don't want to understand since I offered you an alternative treatment using neighbourhoods and you ignored it totally.

 

The beauty of neighbourhoods is that it does not depend upon whether or not there is a smallest real number or a nearest real number or what happens at infinity.

And yet neigherbourhood analysis arrives at the same results as the other approaches you have been arguing about.

 

I read what you said about neighborhoods and infinity.  Then I looked up neighborhoods because I have never heard of them before.  But I still have no idea how it helps me understand anything about my issue, or even how it relates to the discussion.

I am not sure if I am even ready to get into the topic of neighborhoods.  It seems a little or a lot more advanced than what I have learnt so far.

[wtf]

2 hours ago, Boltzmannbrain said:

I meant a space in R².

 

A point in the Cartesian plane is characterized by an ordered pair of real numbers.

But just as with the line, this is misleading. In math there is no geometrical plane. Rather, there are the real numbers, and then ordered pairs of real numbers. And we call the set of ordered pairs of real numbers the "plane" as a geometrical visualization. But there aren't two things, the plane and the set of ordered pairs. There's only one thing, the set of ordered pairs, which we commonly refer to as "the plane," or, when equipped with the Euclidean metric, the Euclidean plane. But it's still just the set of ordered pairs of real numbers. The geometry is just a convenient visualization and way of speaking. 

To emphasize this point, what else can we do with ordered pairs of real numbers? We can define a "funny multiplication" on them, and then they turn into, voilà, the complex numbers C . 

So would we say that the Euclidean plane and the complex numbers are "the same?" Or would we say, rather, that the Euclidean plane and the complex numbers are two different interpretations of the same underlying set of ordered pairs of reals?

I'm mentioning all this because you are insistent in wanting to know how the pairs of real numbers "describe" or "map to" the plane. But in fact, there is no plane. There's only the set of ordered pairs of real numbers, which, as you note, is called \( \mathbb R^2 \).

This is all a bit philosophical or semantic and I don't want to make too much of it. But you may be asking the wrong question. The set of ordered pairs IS the plane, it doesn't map to relate to the plane. If that helps. 

 

 

2 hours ago, Boltzmannbrain said:

If you half the distance an infinite number of times you get 0 don't you? 

 

Yes you get 0 as the limit.

To be clear, in the example of 1/2, 3/4, 7/8, ... we get 1 as the limit. I assume that's what you meant. 

So how does this work? Let's take the example of the sequence 1/2, 1/3, 1/4, 1/5, 1/6, ...

The sequence has 0 as its limit. That means that the terms of the sequence get, and stay, arbitrarily close to 0. That's formalized as the "epsilon" business that people see briefly, in passing, in a beginning calculus course, before getting on to the calculational aspects.

Note that every element of that sequence is NOT zero. This is crucial to the limit concept. The elements of the sequence never "become" zero. Zero is not an element of the sequence. Rather, the sequence has the limit of 0, where we take some pains to formally define what we mean by a limit.

 

 

2 hours ago, Boltzmannbrain said:

I only meant that my intuition has always been that there couldn't be a smallest number (as I believe everyone's intuition is), but now that has changed.

I did not read all the posts in this thread before arriving a few pages in. If you formerly held different beliefs, that might be generating some confusion. But even since you and I have been talking, you seem to believe that there is a largest element of [0,1), and that amounts to the same thing. 

So I cannot square this latest statement with what I've been talking to you about. 

 

 

2 hours ago, Boltzmannbrain said:

Then what you said above cannot be true either about the halving process being carried out an infinite number of times, since n is always finite.

The halving process refers (in our ongoing example of [0, 1)) to the sequence 1/2, 3/4, 7/8, 15/16, ... We can definitely do this infinitely many times, since that is an infinite sequence of rational numbers. There's an element (2^n - 1)/2^n for every natural number n.

The limit of the sequence is 1. But where is the confusion? We have infinitely many terms, each of them finite, and they have a finite limit. No mystery at all.

 

 

2 hours ago, Boltzmannbrain said:

Ok, I understand.  That is very interesting, thanks.  I never took the second semester of the advanced calculus course, so I never got a proper understanding of the more rigorous definition of the Riemann integral.  I only took the typical calculus for learning integrals.

Nobody understands limiting processes in their first calculus course. There's a math major course called Real Analysis in which they nail all this stuff down tight as a drum. But very few people ever take that course. So a lot of people, especially technical people like engineers and physicists, are great at calculus but were never exposed to the subtleties of the nature of the real numbers and limiting processes. Perfectly normal. 

 

 

2 hours ago, Boltzmannbrain said:

Then how can we get 0 when calculating the widths of the partitions over n as n goes to infinity?  What finite n value would allow this?

As a limit. I hope you see the ongoing theme of this post!

In the Riemann integral, we take finer and finer partitions. Say the partitions have successive widths 1/2, 1/3, 1/4, 1/5, ... Every partition has nonzero width. The limit of the widths is zero. And in fact we do NOT require a "zero width" when defining the Riemann integral. Rather, for each partition, we calculate the corresponding Riemann sum, the sum of the areas of the finitely many rectangles in that partition; and we define the integral as the limit of the sequence of Riemann sums.

As you can see, everything depends on a proper understanding of the limit concept. 

 

 

2 hours ago, Boltzmannbrain said:

I am interested in the integral because of what we are discussing above.  It seems like that will help me understand the ideas of density, infinity vs limit to infinity, etc. from a different perspective, namely the integral (well to be more specific, the 1 dimensional aspect of the integral).

That's a little word-salady to me. What we were discussing above is whether there's a largest number in [0,1), which it turns out there isn't. So whatever interest you about the Riemann integral may have been explained before I joined the thread. Feel free to catch me up. 

And what is the "one dimensional aspect of the integral?" I don't know what you mean. 

 

 

2 hours ago, Boltzmannbrain said:

I would think so too.  Except when I think about that example I gave earlier about the segment not fitting because of one extra point, I am left confused and very curious.

I thought I explained it. You are correct that [0,1] and [1,2] have a point in common, and if you tried to rotate them past each other they'd indeed collide at the point 1.

You are wondering how they can collide if the point 1 has zero length. I tried to analogize this with a Newtonian point mass. I said that even though 1 has zero length, zero dimension, zero volume, it can "still pack a punch."

Did this analogy not resonate with you, even a little? Maybe it's kind of strained, I did admit that up front. 

But in your rotating visualization, the line segments will indeed collide at the point 1, simply because 1 is an element of each segment. 

 

 

Edited March 15, 2023 by wtf

[studiot]

19 hours ago, Boltzmannbrain said:

I read what you said about neighborhoods and infinity.  Then I looked up neighborhoods because I have never heard of them before.  But I still have no idea how it helps me understand anything about my issue, or even how it relates to the discussion.

I am not sure if I am even ready to get into the topic of neighborhoods.  It seems a little or a lot more advanced than what I have learnt so far.

 

So how am I meant to take this ?

You don't know anything about neighbourhoods but you don't want to consider them ?

Why not ?

Why is this not inconsiderate of others who are trying to help you ?

 

As far as I can see you are having trouble reconciling the concept of  infinity and infinite processes with your half completed half remembered past studies.

Further you keep intorducing infinity and things which are not numbers or even necessarily to do with numbers.

On 3/13/2023 at 3:54 PM, Boltzmannbrain said:

But that is how I am using an infinite number.  Since there are an infinite number of numbers/points or decreasing subsets from 0 to 1, then it would seem fair to use an infinite number where it is relevant.

 

19 hours ago, Boltzmannbrain said:

If you half the distance an infinite number of times you get 0 don't you

 

19 hours ago, Boltzmannbrain said:

Then how can we get 0 when calculating the widths of the partitions over n as n goes to infinity?  What finite n value would allow this?

 

19 hours ago, Boltzmannbrain said:

I am interested in the integral because of what we are discussing above.  It seems like that will help me understand the ideas of density, infinity vs limit to infinity, etc. from a different perspective, namely the integral (well to be more specific, the 1 dimensional aspect of the integral).

 

As soon as you start talking about integrals you have left the realm of numbers - real or otherwise - and entered the realm of functions.

Functions are not numbers and numbers are not functions.

In fact the Riemann integral is a mapping from the space of functions to the space of numbers  (please note the correct use of the word space here)

Not only that it is a linear mapping called a functional.

 

Be that as it may here is one way in which neighbourhoods help avoid infinity.

There are different types of neighbourhood defined.

A simple neighbourhood of a general point p which includes the point p itself

and a reduced neighbourhod of p which does not include the point p itself.

By using the latter we may avoid infinity if we define a reduced neighbourhood of p as ∞ as being {x: x > c} where C is some chosen number.

Do you understand this notation ?

It means the set of all numbers greater than some C.

The implications of this are huge.

It means that it does nor matter whether there is an infinity or not - we only have to 'approach it' or 'tend to it'.

Equally useful mileage can be obtained from the reduced neighbourhood of a real number point p, since it must include any 'nearest real number' - if such a thing exists.
But we can still proceesd with out analysis if it does not.

Also, and back to integrations and functions, we can use neighbourhoods directly with functions.
We cannot so use the other derivations without modification.

 

What's not to like ?

Here is a link to a good article about intervals and neighbouthoods.

Note they use 'a' for the point, p where i have use 'p'.

https://math.fel.cvut.cz/en/mt/txtb/1/txe3ba1a.htm

[Boltzmannbrain]

On 3/15/2023 at 2:43 PM, wtf said:

A point in the Cartesian plane is characterized by an ordered pair of real numbers.

But just as with the line, this is misleading. In math there is no geometrical plane. Rather, there are the real numbers, and then ordered pairs of real numbers. And we call the set of ordered pairs of real numbers the "plane" as a geometrical visualization. But there aren't two things, the plane and the set of ordered pairs. There's only one thing, the set of ordered pairs, which we commonly refer to as "the plane," or, when equipped with the Euclidean metric, the Euclidean plane. But it's still just the set of ordered pairs of real numbers. The geometry is just a convenient visualization and way of speaking. 

To emphasize this point, what else can we do with ordered pairs of real numbers? We can define a "funny multiplication" on them, and then they turn into, voilà, the complex numbers C . 

So would we say that the Euclidean plane and the complex numbers are "the same?" Or would we say, rather, that the Euclidean plane and the complex numbers are two different interpretations of the same underlying set of ordered pairs of reals?

I'm mentioning all this because you are insistent in wanting to know how the pairs of real numbers "describe" or "map to" the plane. But in fact, there is no plane. There's only the set of ordered pairs of real numbers, which, as you note, is called R2 .

This is all a bit philosophical or semantic and I don't want to make too much of it. But you may be asking the wrong question. The set of ordered pairs IS the plane, it doesn't map to relate to the plane. If that helps. 

 

 

That makes sense.  It is what I was thinking.

 

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Yes you get 0 as the limit.

To be clear, in the example of 1/2, 3/4, 7/8, ... we get 1 as the limit. I assume that's what you meant. 

So how does this work? Let's take the example of the sequence 1/2, 1/3, 1/4, 1/5, 1/6, ...

The sequence has 0 as its limit. That means that the terms of the sequence get, and stay, arbitrarily close to 0. That's formalized as the "epsilon" business that people see briefly, in passing, in a beginning calculus course, before getting on to the calculational aspects.

Note that every element of that sequence is NOT zero. This is crucial to the limit concept. The elements of the sequence never "become" zero. Zero is not an element of the sequence. Rather, the sequence has the limit of 0, where we take some pains to formally define what we mean by a limit.

 

 

Okay, that makes sense.

 

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I did not read all the posts in this thread before arriving a few pages in. If you formerly held different beliefs, that might be generating some confusion. But even since you and I have been talking, you seem to believe that there is a largest element of [0,1), and that amounts to the same thing. 

So I cannot square this latest statement with what I've been talking to you about. 

 

 

I only meant that my intuition tells me recently that there is a smallest real.

 

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The halving process refers (in our ongoing example of [0, 1)) to the sequence 1/2, 3/4, 7/8, 15/16, ... We can definitely do this infinitely many times, since that is an infinite sequence of rational numbers. There's an element (2^n - 1)/2^n for every natural number n.

 

I think this all starting to sink in.  

 

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Yes you get 0 as the limit.

To be clear, in the example of 1/2, 3/4, 7/8, ... we get 1 as the limit. I assume that's what you meant. 

So how does this work? Let's take the example of the sequence 1/2, 1/3, 1/4, 1/5, 1/6, ...

The sequence has 0 as its limit. That means that the terms of the sequence get, and stay, arbitrarily close to 0. That's formalized as the "epsilon" business that people see briefly, in passing, in a beginning calculus course, before getting on to the calculational aspects.

Note that every element of that sequence is NOT zero. This is crucial to the limit concept. The elements of the sequence never "become" zero. Zero is not an element of the sequence. Rather, the sequence has the limit of 0, where we take some pains to formally define what we mean by a limit.

 

 

Okay, this makes sense too.

 

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As a limit. I hope you see the ongoing theme of this post!

In the Riemann integral, we take finer and finer partitions. Say the partitions have successive widths 1/2, 1/3, 1/4, 1/5, ... Every partition has nonzero width. The limit of the widths is zero. And in fact we do NOT require a "zero width" when defining the Riemann integral. Rather, for each partition, we calculate the corresponding Riemann sum, the sum of the areas of the finitely many rectangles in that partition; and we define the integral as the limit of the sequence of Riemann sums.

As you can see, everything depends on a proper understanding of the limit concept. 

 

 

I see.  Thanks, I forgot about that.

 

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That's a little word-salady to me. What we were discussing above is whether there's a largest number in [0,1), which it turns out there isn't. So whatever interest you about the Riemann integral may have been explained before I joined the thread. Feel free to catch me up. 

 

I was only discussing it with you.  I forgot/misunderstood the integral process.

 

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I thought I explained it. You are correct that [0,1] and [1,2] have a point in common, and if you tried to rotate them past each other they'd indeed collide at the point 1.

You are wondering how they can collide if the point 1 has zero length. I tried to analogize this with a Newtonian point mass. I said that even though 1 has zero length, zero dimension, zero volume, it can "still pack a punch."

Did this analogy not resonate with you, even a little? Maybe it's kind of strained, I did admit that up front. 

But in your rotating visualization, the line segments will indeed collide at the point 1, simply because 1 is an element of each segment. 

 

 

This seems interesting to me in that adding one real number is significant geometrically, but I don't know right how relevant it is to this discussion.

 

Thank you very much for your help and incredible patience

 

 

 

On 3/16/2023 at 8:16 AM, studiot said:

 

So how am I meant to take this ?

You don't know anything about neighbourhoods but you don't want to consider them ?

Why not ?

Why is this not inconsiderate of others who are trying to help you ?

 

Sorry, I really appreciate your help, but I wasn't making the connection with what you were saying and my issue.  Also, I did not have enough time to figure it out.

I think I have finally understood this whole issue from my OP completely.   

Thank you for your help!

Edited March 18, 2023 by Boltzmannbrain

[studiot]

1 hour ago, Boltzmannbrain said:

I think I have finally understood this whole issue from my OP completely.   

Thank you for your help!

Well congratulations, I really hope so. +1

 

If I might offer a final piece of advice.

Next time try to focus on what yoy need to know for one aspect of analysis at a time.

I know everyhting joins up in the end but that is only when you have covered everything.

Trust you helpers to help narrow this down with you instead of rambling about all over the whole subject of analysis or functions of a real variable or whatever you wish to call the subject.

Edited March 18, 2023 by studiot

[joigus]

On 3/12/2023 at 10:51 AM, studiot said:

Impressive and patient replies.  +1

Agreed.