# Concerns about the geometry of the real number line

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36 minutes ago, Boltzmannbrain said:

I want to take a line that is in a real space and divide it until there is nothing left, sort of like we do with Reiman integrals as n goes to infinity.  Is this possible?

Subject to the comment Genady has already made about Reiman integrals, yes you can and The method is nothing like you seem to envisage and nothing to do with Reiman integrals.

What you are seeking is the method of partitioning a set so that every partition contains exactly one member of the set and there are no empty partitions.

This is called the method of Dedekind cuts.

But it does not lead to a smallest real number,

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6 hours ago, studiot said:

One or two debatable points there but your heart is in the right place.

+1 for encouragement

Whilst @Genady is doing such a good job offering straightforward textbook details I think it is worth looking deeper into the number line concept.

I will readily grant you that the number line is a very attractive way of presenting aspects of the properties of numbers.

But what we are really doing is as in fig1 putting numbers in one-to-one correspondence with a straight line and hoping that other properties are then transferable.

But also as in fig 1 it is easy to see that the actual length of the line is irrelevant - it works for all straight lines, regardless of theier length.

Radical conclusion : You can establish one-to-one correspondence for the whole set of numbers with the shortest possible straight line, as well as the longest possible.

Question : How does this affect you proposed distance function ?

Surely the distances apart of any two numbers will depend upon the length of you line.

But who says the line must be straight ?

Fig 2 shows how to use a spiral to place the numbers.

This has the interesting property of being able to cover the entire plane.

Question : How does this affect you proposed distance function ?

As can be seen in  fig 2   - the numbers 2,4 and 6 are all equally close to the number 1, whereas the numbers 3 and 5 are further away.

But does the line have to have a definite geoemtric shape ?  So far I have discussed straight and spiral.

Surely all that is required is that it must not cross itself.

Question : How does this affect you proposed distance function ?  Can you see why it must not cross ?

Fig3 shows one such semi random arrangement. There are an infinite count of such non crossing arangements.

What about the numbers themselves. There is nothing in basic set theory to say that they have to appear in the set in any particular configuration.

Fig 4 shows a random positioning of the numbers which is perfectly permissible.

Question : How does this affect you proposed distance function ?

Clearly 4 is the closest number to 1.

So that you can have them is value order you have to add further structure to your set of numbers so that you can introduce a 'well ordering' of the set.

Thanks but I do not understand what this has to do with me trying to find a next real number.  These are natural numbers.  Do the lines that connect these natural numbers exist as infinitely many points, or are they just there to show me what number comes next.  I am really confused.

45 minutes ago, joigus said:

This is probably the best starting point for this question --no pun intended.

If x, y are in R, and x y. Then,

Either x>y or x<y. Assume x<y.

A property of the real numbers is there always exists a z in R such that x<z<y if x and y are different. So there isn't such a thing as 'next number' in R.

Is this an axiom of the reals or an implication from other axioms?

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3 minutes ago, Boltzmannbrain said:

Thanks but I do not understand what this has to do with me trying to find a next real number.  These are natural numbers.  Do the lines that connect these natural numbers exist as infinitely many points, or are they just there to show me what number comes next.  I am really confused.

Is this an axiom of the reals or an implication from other axioms?

I would call it a lemma or proposition --small theorem-- that's easy to prove.

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So, now you are talking about a different 'line', not the one we have defined earlier in real vector space as per the linked video? If so, we need a new definition.

I think you are making this way harder than it has to be.  I am trying to consider a line like that would resemble the real number line in whatever part of math this exists.

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Riemann integral does not deal with lines, it deals with numbers.

That's why I said "sort of".  Your misinterpretations are getting quite painful.

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No, they cannot be real numbers. Points are not numbers and numbers are not points.

Can't we place numbers in a line segment that would have a property of being a point?

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R is a set of real numbers. There are no lines in this set, only numbers.

But it "forms" a continuum don't they?  A line in R would seem to have similar properties.

I thought that I have already shown to you earlier that there is no 'smallest real number', haven't I?

Here is what I told you in a previous post,

I am just really interested in this subject.  I don't actually think that I will find a contradiction, but I want to just go on the endeavour anyways.  If I don't find one (contradiction or smallest number), then I am just happy to learn more about infinity and its properties.

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19 minutes ago, Boltzmannbrain said:

I think you are making this way harder than it has to be.  I am trying to consider a line like that would resemble the real number line in whatever part of math this exists.

That's why I said "sort of".  Your misinterpretations are getting quite painful.

Can't we place numbers in a line segment that would have a property of being a point?

But it "forms" a continuum don't they?  A line in R would seem to have similar properties.

Here is what I told you in a previous post,

I am just really interested in this subject.  I don't actually think that I will find a contradiction, but I want to just go on the endeavour anyways.  If I don't find one (contradiction or smallest number), then I am just happy to learn more about infinity and its properties.

I just try to replace vague words like "same way", "sort of", "forms" (in parentheses) by clarity. I don't think that vague talk is a fruitful endeavor. But if it is not what you're looking for, then fine.

52 minutes ago, Boltzmannbrain said:

Is this an axiom of the reals or an implication from other axioms?

Sorry to say, but your going in circles is getting quite painful. It has been already covered and referred to.

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1 hour ago, Boltzmannbrain said:

Thanks but I do not understand what this has to do with me trying to find a next real number.

!

Moderator Note

Three pages in and you still aren't listening to what's being said. Just because you can't believe there's no next real number doesn't make it so, and your incredulity alone means nothing. It's too frustrating to be in a discussion with someone who is wrong about something but refuses to listen to all the folks trying to help. Every single poster has given you a different, reasonable way to deal with this problem, and you've rejected every one.

You've admitted that your original argument is dead. Most of your concerns are nebulous and involve how an explanation makes you "feel". At a certain point, arguments like these become soapboxing with no reasoning to support them, and nobody wants to be in a discussion like that. Decide whether or not you have anything to support your explanation, or perhaps that you were wrong and need to review what's already been given to you.

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Posted (edited)
On 3/5/2023 at 7:46 PM, Boltzmannbrain said:

I think you are making this way harder than it has to be.

This statement I can agree with.

So here is a simple observation.

The ancient Greeks realised that

If r is a positive number, real or otherwise, then r+1 is a larger number, and (r+1)+1 is larger still

and so on.

So there is no largest number.

Now we also know that if we take reciprocals

1/r is greater than 1/(r+1) which is greater than 1/((r+1)+1)

and so on.

Since there is no largest number there can be no smallest number either.

Sweet dreams everybody.

Edited by studiot
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On 3/5/2023 at 7:26 PM, Boltzmannbrain said:

Thanks but I do not understand what this has to do with me trying to find a next real number.  These are natural numbers.  Do the lines that connect these natural numbers exist as infinitely many points, or are they just there to show me what number comes next.  I am really confused.

Is this an axiom of the reals or an implication from other axioms?

The real numbers can have the" next " number?

The answer to this question is a resounding no. Real numbers are already infinite, so it's impossible for them to have the "next" number - after all, they don't even know what that would be! It's like asking an endless ocean if it can contain one more drop of water; the answer will always be no!

Intervals on the real number line are an important concept in math, but they don't have to be complicated!  For example, if we look at a number line from 0 to 10, then (2, 8) is an interval that includes all numbers between 2 and 8 (but not including either of those two numbers). Similarly, [5.5 , 9] is another interval that contains all real numbers starting from 5.5 up until 9 - simple as can be!

Intervals on the real number line are like a date night for math -- nothing too serious or committed, just a pleasant distraction from the day-to-day of dealing with numbers that just won't behave.

They divide up the real number line into manageable chunks so there's no more guessing whether you should be adding, subtracting, multiplying or dividing.

Plus it's got some pretty nifty consequences for graphing equations (no velocity limit here!).   any more questions?

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Posted (edited)
5 hours ago, Mc2509 said:

The real numbers can have the" next " number?

The answer to this question is a resounding no.

so it is a pity your reasoning is incorrect.

5 hours ago, Mc2509 said:

Real numbers are already infinite, so it's impossible for them to have the "next" number

furthermore  numbers don't 'know' anything.

5 hours ago, Mc2509 said:

after all, they don't even know what that would be!

5 hours ago, Mc2509 said:

It's like asking an endless ocean if it can contain one more drop of water; the answer will always be no!

Edited by studiot
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6 hours ago, Mc2509 said:

It's like asking an endless ocean if it can contain one more drop of water; the answer will always be no!

One has to be careful even with the natural numbers. The ocean of natural numbers admits arbitrarily many more drops!

The real numbers are even more counterintuitive.

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1 hour ago, joigus said:

The real numbers are even more counterintuitive.

Moreover, we never observe or measure real numbers. Unlike natural numbers - we can count - and rational numbers - we can break things.

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Posted (edited)
On 3/4/2023 at 6:22 PM, Boltzmannbrain said:

I believe distance can be defined by Euclidean distance which uses Pythagorean theorem.

Can it ?

Let us try some examples.

Compare the distance between 1 and 2 and between 4 and 5 by this definition.

Distance 1,2  = √(12 + 22)  =  (1 +4) √5

Distance 4,5 = √(42 + 52) = (16 + 25) = √41

Is this difference acceptable ?

If you read my Wiki reference on Metric spaces you would see that there are three different common definitions of distance and the euclidian one is not appropriate here.

I have already mentioned the method of Dedikind.

This used to be the preferred route to understanding why we can keep chopping up the distance between any two real numbers.

The old way was to start with counting numbers, (whose distance betweeen is not infinitely divisible)

show why these are not enough ie there are more whole number than counting numbers  to get the integers

Then progress to show that there rational numbers give us still more numbers the distance between becoming infinitely divisible.

But also that there are gaps in between which means that the rational numbers are not continuous.

Finally we come to the real numbers which are continuous in that there are no gaps at all such that the real numbers possess the property we call completeness as well as being infinitely divisible like the rationals which are not complete.

Being complete and without gaps means that that there are no further numbers to acount for.

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Because of some frustration that was expressed, I was going to bail on this thread.

But I thought I would continue because I am just too interested in this subject. And I would like to explain myself better.

One thing I would like to say to everyone coming into this thread, which I should've been more clear on in the OP, is that this topic is still being researched.  So there probably won't be a perfect and irrefutable answer to the general concern in the OP.

Having said that, I am here to learn, teach, discuss, etc.  I am not here to frustrate anyone, or troll.  Please give me a chance to dig into this with you all.

7 hours ago, studiot said:

If r is a positive number, real or otherwise, then r+1 is a larger number, and (r+1)+1 is larger still

and so on.

So there is no largest number.

I was told a long time ago by my math professor that aleph null (the cardinal size of the naturals/rationals or the "smallest infinity") can actually be used as a number.  So in set theory, I think we can actually say that there is a number larger than any r + 1 or any natural number.

It is interesting to me that when aleph null is used as a number in the case of 2^(aleph null), it equals the next largest infinity being the set of the reals.

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Can it ?

Let us try some examples.

Compare the distance between 1 and 2 and between 4 and 5 by this definition.

Distance 1,2  = √(12 + 22)  =  (1 +4) √5

Distance 4,5 = √(42 + 52) = (16 + 25) = √41

Is this difference acceptable ?

This link shows the procedure,  Euclidean distance - Wikipedia

Quote

If you read my Wiki reference on Metric spaces you would see that there are three different common definitions of distance and the euclidian one is not appropriate here.

I have already mentioned the method of Dedikind.

This used to be the preferred route to understanding why we can keep chopping up the distance between any two real numbers.

The old way was to start with counting numbers, (whose distance betweeen is not infinitely divisible)

show why these are not enough ie there are more whole number than counting numbers  to get the integers

Then progress to show that there rational numbers give us still more numbers the distance between becoming infinitely divisible.

But also that there are gaps in between which means that the rational numbers are not continuous.

Finally we come to the real numbers which are continuous in that there are no gaps at all such that the real numbers possess the property we call completeness as well as being infinitely divisible like the rationals which are not complete.

Being complete and without gaps means that that there are no further numbers to acount for.

I am still going through the link you gave me.  There is quite a bit of information there to consider.  I am also trying to think more about how the Dedikind cut comes into this.

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17 minutes ago, Boltzmannbrain said:

I was told a long time ago by my math professor that aleph null (the cardinal size of the naturals/rationals or the "smallest infinity") can actually be used as a number.  So in set theory, I think we can actually say that there is a number larger than any r + 1 or any natural number.

What sort of professor was this fellow  and what level mathematics was he teaching ?

If and only if ${\aleph _0}$ is a number then you can add 1 to it.

That is the most basic property of any number system there is.

However I suggest you forget the alephs because none of them are real numbers, unlike the numbers I have been using.
I did not say natural number, 1 ,2, 3, 4, 5 , 6 , 7 ,8 ,8, 0 are all real numbers.

I said several times I am using these because it is simpler. Using 1.05788 for instance sould be done but obscures the real point of a statement.

25 minutes ago, Boltzmannbrain said:

This link shows the procedure,  Euclidean distance - Wikipedia

Yes and it states quite clearly that this you have to have cartesian coordinates to execute this procedure.

Numbers do not have cartesian coordinates.

On the real number line, the value of the number is the same as its distance from ther cartesian origin of the geometrical object we call the real number line.

Talking of properties have you thought about reviewing your starting point for this ?

You need to distinguish very carefully between the value of a number and some distance you associate with it, because your question is about the value of that number not any associated distance.

Nearly two and half thousand years ago it was realised that numbers went on forever and they further realised that there were additional numbers (which they tried to hide) that did not fit into their system.

It was not until 1872 when Dedekind published his theorem or axiom that this issue was properly resolved.

I have also mentioned the upper and lower bound theorems.

Cantor's approach to the issue was one of synthesis - build up the whole from small parts.

Dedekind approached from the other point of view - analysis - break down the whole into small parts.

His axiom shows the difference between the rational numbers and the real numbers.

The real numbers include all the numbers that do not fit into the rational number system - because they are irrational.

He proved, using Cantor's set theory, that the real number system is 'complete'; that is there is a place for all numbers in it.

The rational number system by contrast is incomplete, known as already noted for a very long time.

You can carry on the many of the same processes in either the rational or real number systems, in particular the infinite processes of tending to infinity or the infinitesimal.

But there is a second property that makes the difference.

Take for instance the square root of 2, probably the most common irrational number.

Schoolboys are shown the quick and dirty algebraic proof that it is irrational.

But they rarely probe further in to the meaning of this which is that the square root of 2 is not a member of the set of rational numbers.
You can never get there by any process infinite or finite since the number just does not exist in the rationals since it has no place there.

Infinity by itself is not enough.

It is very illuminating to use Dedekind to develop why this is and then go on to the second required property, that of completeness.

We can do that together, if you like.

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2 hours ago, Boltzmannbrain said:

this topic

2 hours ago, Boltzmannbrain said:

general concern

Frankly, I don't know now what topic/concern is/are discussed. Is it something definite?

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2 hours ago, studiot said:

What sort of professor was this fellow  and what level mathematics was he teaching ?

Fortunately I attended a good university.  He got his PhD from Princeton.  It was the advanced calculus course.

2 hours ago, studiot said:

If and only if 0 is a number then you can add 1 to it.

That is the most basic property of any number system there is.

However I suggest you forget the alephs because none of them are real numbers, unlike the numbers I have been using.
I did not say natural number, 1 ,2, 3, 4, 5 , 6 , 7 ,8 ,8, 0 are all real numbers.

I said several times I am using these because it is simpler. Using 1.05788 for instance sould be done but obscures the real point of a statement.

I thought r was suppose to mean a real number.  In any case, my point was to bring up an example to remind us of part of the scope of set theory.

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Yes and it states quite clearly that this you have to have cartesian coordinates to execute this procedure.

Ok, but where did I say that we didn't need cartesian coordinates?

Quote

Numbers do not have cartesian coordinates.

On the real number line, the value of the number is the same as its distance from ther cartesian origin of the geometrical object we call the real number line.

That makes sense.

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Talking of properties have you thought about reviewing your starting point for this ?

You need to distinguish very carefully between the value of a number and some distance you associate with it, because your question is about the value of that number not any associated distance.

Nearly two and half thousand years ago it was realised that numbers went on forever and they further realised that there were additional numbers (which they tried to hide) that did not fit into their system.

It was not until 1872 when Dedekind published his theorem or axiom that this issue was properly resolved.

Interesting!

Quote

I have also mentioned the upper and lower bound theorems.

Cantor's approach to the issue was one of synthesis - build up the whole from small parts.

Dedekind approached from the other point of view - analysis - break down the whole into small parts.

His axiom shows the difference between the rational numbers and the real numbers.

The real numbers include all the numbers that do not fit into the rational number system - because they are irrational.

He proved, using Cantor's set theory, that the real number system is 'complete'; that is there is a place for all numbers in it.

The rational number system by contrast is incomplete, known as already noted for a very long time.

You can carry on the many of the same processes in either the rational or real number systems, in particular the infinite processes of tending to infinity or the infinitesimal.

But there is a second property that makes the difference.

Take for instance the square root of 2, probably the most common irrational number.

Schoolboys are shown the quick and dirty algebraic proof that it is irrational.

But they rarely probe further in to the meaning of this which is that the square root of 2 is not a member of the set of rational numbers.
You can never get there by any process infinite or finite since the number just does not exist in the rationals since it has no place there.

Infinity by itself is not enough.

It is very illuminating to use Dedekind to develop why this is and then go on to the second required property, that of completeness.

We can do that together, if you like.

Sure that sounds useful and interesting.

Frankly, I don't know now what topic/concern is/are discussed. Is it something definite?

I do not understand how it is logical to have no next number on a line segment in a real space.  I am hoping to explore the implications of removing a number at the end of the line segment.  Why does the end of the line segment no longer have an end number?  The geometrical result seems illogical to me.

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11 minutes ago, Boltzmannbrain said:

I do not understand how it is logical to have no next number on a line segment in a real space.  I am hoping to explore the implications of removing a number at the end of the line segment.  Why does the end of the line segment no longer have an end number?  The geometrical result seems illogical to me.

OK, let's explore it.

Let's assume that a line segment always has an end number. What happens when we remove that number? There are two possibilities:

1. It is impossible to remove a number from a segment. It is only possible to remove some open range of numbers, however small it is, but not one number only. One individual number is not removable.

Or,

2. If one number is removed from a segment, what's left is not a segment anymore, but some other kind of object.

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OK, let's explore it.

Let's assume that a line segment always has an end number. What happens when we remove that number? There are two possibilities:

1. It is impossible to remove a number from a segment. It is only possible to remove some open range of numbers, however small it is, but not one number only. One individual number is not removable.

Or,

2. If one number is removed from a segment, what's left is not a segment anymore, but some other kind of object.

Is one of these possibilities correct?

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3 minutes ago, Boltzmannbrain said:

Is one of these possibilities correct?

I think, the no. 1.

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Posted (edited)
2 hours ago, Boltzmannbrain said:

I do not understand how it is logical to have no next number on a line segment in a real space.  I am hoping to explore the implications of removing a number at the end of the line segment.  Why does the end of the line segment no longer have an end number?

Consider this example.

Consider the closed unit interval $[0,1]$ consisting of all the real numbers between 0 and 1, inclusive. Suppose that we delete 1 from the right-hand end, leaving us with $[0,1) = \{x \in \mathbb R : 0 \leq x < 1\}$. That is, we are considering the set of all real numbers greater than or equal to 1, and strictly less than 1.

Now I claim that this set has no largest number, or no right-handed endpoint. How do I know that? Well, suppose you claim that, say, 3/4 is the largest. I'd just note that 7/8 is strictly larger and still strictly smaller than 1. And what if you say ok then how about 7/8? I'd note that 15/16 is strictly between 7/8 and 1.

You can see that no matter what number you claim is the new right-hand endpoint, it can't be; because there is always a number halfway between your number and 1 that is strictly greater than your number, and strictly less than 1.

Do you see that the interval $[0,1)$ has no right-hand endpoint?

On 3/5/2023 at 10:53 AM, studiot said:

What you are seeking is the method of partitioning a set so that every partition contains exactly one member of the set and there are no empty partitions.

If I have any set $X$ I can partition it into a collection of singletons as

$\displaystyle X = \bigcup_{x \in X} \{x\}$

I've partitioned an arbitrary set such that "every partition contains exactly one member of the set and there are no empty partitions." That's not a Dedekind cut, it's just the trivial partitioning of a set into the union of its singletons.

A Dedekind cut consists of a two-set partition of the rationals such that all the elements of one element of the partition are strictly less than all the elements of the other. For example defining sqrt(2) as the partition of the rationals into all the rationals whose squares are respectively less than 2 and greater than 2.

Dedekind cuts are not particularly helpful to resolving the OP's concern about "next" real numbers; but I haven't read the entire thread so perhaps they're relevant to some other aspect.

Edited by wtf
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1 hour ago, wtf said:

Consider this example.

Consider the closed unit interval [0,1] consisting of all the real numbers between 0 and 1, inclusive. Suppose that we delete 1 from the right-hand end, leaving us with [0,1)={xR:0x<1} . That is, we are considering the set of all real numbers greater than or equal to 1, and strictly less than 1.

Now I claim that this set has no largest number, or no right-handed endpoint. How do I know that? Well, suppose you claim that, say, 3/4 is the largest. I'd just note that 7/8 is strictly larger and still strictly smaller than 1. And what if you say ok then how about 7/8? I'd note that 15/16 is strictly between 7/8 and 1.

You can see that no matter what number you claim is the new right-hand endpoint, it can't be; because there is always a number halfway between your number and 1 that is strictly greater than your number, and strictly less than 1.

Do you see that the interval [0,1) has no right-hand endpoint?

This argument definitely makes sense.  But I wonder if it is just a method that just doesn't work in finding a next number.

Anyway, I have to give the counter-argument that is nagging at me.  We have a set of numbers inclusively from 0 until 1, [0, 1].  The numbers increase in order to the right.  There is a final number to the set, which is 1.  It is furthest to the right, so logically it would be the greatest of the other numbers.  We take away the number 1 from the set.  Okay so far this all makes complete sense.  It is all very intuitive, and there is nothing strange or even interesting happening.

Now I am told that this set of increasing numbers no longer has a greatest number or an end.  The number 1 seemed to have had some sort of special property that the other digits don't have.   It can be at the end as well as the set's greatest number.  Why can the number 1 do this and no other number?  This does not seem logical to me, so I must be missing something.

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Posted (edited)
3 hours ago, Boltzmannbrain said:

This argument definitely makes sense.

Thank you, glad it was helpful. After I wrote my post I went back and read through this entire thread, and I see I haven't actually said anything that hasn't already been said, so credit to all the others here before me.

3 hours ago, Boltzmannbrain said:

But I wonder if it is just a method that just doesn't work in finding a next number.

It's much more than that. It's a definitive proof that $[0,1)$ has no largest element. Because if you claim that $x$ is the largest element, I can just note that $x + \frac{1 - x}{2}$ is strictly between $x$ and $1$. I hope you can see that. This point was made by @Genady on page 1 of this thread. He noted that if p and q are real numbers then (p + q)/2 is a real number strictly between them. It's essentially the same argument that I just gave.

3 hours ago, Boltzmannbrain said:

Anyway, I have to give the counter-argument that is nagging at me.  We have a set of numbers inclusively from 0 until 1, [0, 1].  The numbers increase in order to the right.  There is a final number to the set, which is 1.  It is furthest to the right, so logically it would be the greatest of the other numbers.  We take away the number 1 from the set.  Okay so far this all makes complete sense.  It is all very intuitive, and there is nothing strange or even interesting happening.

Well it IS strange and interesting, that's why you asked! It is counterintuitive that given a real number, there is not a "next" point; and that if a line segment is missing its right endpoint, then it can't contain a largest number. Many people are confused about this, and it takes some understanding of the real numbers to correct our false intuitions.

If it helps, think of the real numbers as maple syrup, or an infinitely stretchy rubber band; and not as a string of bowling balls. The real numbers are not lined up next to each other. Between any two real numbers are infinitely many more.

3 hours ago, Boltzmannbrain said:

Now I am told that this set of increasing numbers no longer has a greatest number or an end.

Indeed, I just showed you a proof. Any candidate real number in [0, 1) that you might claim is the largest isn't. That's because you can take half the distance between your number and 1, and add it to x, to get a number strictly between your number and 1.

3 hours ago, Boltzmannbrain said:

The number 1 seemed to have had some sort of special property that the other digits don't have.   It can be at the end as well as the set's greatest number.  Why can the number 1 do this and no other number?  This does not seem logical to me, so I must be missing something.

It's a boundary point. That's one of the topological notions mentioned by @studiot on page 1.

A boundary point has the property that any interval surrounding it necessarily contains points in the original interval, as well as points outside it. If you look at the closed unit interval [0,1] (remember that contains both endpoints) the right endpoint is indeed special. Any interval that contains it, such as, say, (.99, 1.01), contains points that are in [0,1] and also points not in [0,1]. That's the definition of a boundary point.

A point in inside the interval, like 0.5, does not have that property. There are small intervals around 0.5 that are entirely contained within [0,1]. The name for that kind of point is interior point.

So yes, boundary points are special. When you remove one of the endpoints from an interval, that end of the interval becomes open, and gets "fuzzy" if you think of it that way.

It's sometimes more geometrically enlightening to think of it in two dimensions. Consider the unit disk consisting of all the points within 1 unit of the origin. If we include the boundary, we have the set $\{(x,y) : x^2 + y^2 \leq 1\}$; and if we exclude the boundary, we have $\{(x,y) : x^2 + y^2 < 1\}$.

You can see that if we omit the boundary, for any point inside the disk, there's another point strictly between that point and the boundary. If we omit the boundary of the disk, there are no points that are "closest to the boundary." It gets fuzzy out there.

Here's a picture I found online. You can see that a point on the boundary is characterized by the fact that any (blue and pink) disk we draw around it contains points of the original disk and points outside the disk. And any interior point is characterized by our ability to find a (blue) disk containing that point that's entirely within the original disk.

Edited by wtf
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Removing a boundary point is akin removing surface from water. It is possible to remove a very thin layer of water from the surface, but it is not possible to remove the surface and to leave the water without a surface.

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6 hours ago, Boltzmannbrain said:

This argument definitely makes sense.  But I wonder if it is just a method that just doesn't work in finding a next number.

The point is that you need to set the stage you are working on.

That is the point of all the different offereings about open and closed sets, squences and so on.

In what circumstances are you looking for a next number ?

My stage is concerned with what are numbers? and what properties do we want them to have ?

11 hours ago, Boltzmannbrain said:

Fortunately I attended a good university.  He got his PhD from Princeton.  It was the advanced calculus course.

Thank you that I hope that is useful information to everybody.

I expect he alsotold you that each Aleph is the cardinality of a particular infinite set of numbers.
And perhaps he also made the point that it is not necessary for the cardinal number of the set to be an element of that set.

I can be for finite sets, but it can't be for infinite sets.

11 hours ago, Boltzmannbrain said:

I thought r was suppose to mean a real number.  In any case, my point was to bring up an example to remind us of part of the scope of set theory.

So often we all read something into what someone else has said, that was just not there.

I do it, though I try to guard against it.

In this case I specifically said the opposite

On 3/6/2023 at 9:30 PM, studiot said:

If r is a positive number, real or otherwise,

The thing about the rational numbers is that they have the same property we are examining as the reals, which is called density or denseness.

Density should not be confused with completeness.
In colloquial terms density means that between any two rational numbers we can find another rational number.
This is an easy 3 line algebraic proof/demonstration if you want it.

So the rationals also have this property of no nearest neighbour, assuming you have decided what you mean by 'nearest'.

So what is the difference between the rationals and the reals ?

Understanding this difference is the great step forward that Dedekind took when he published

Stetigkeit und irrationale Zahlen, Braunschweig, in 1872

Yes WTF is correct in that Dedekind Cuts (Schnitt) divides the entire set of numbers into two subsets, but there is more to Dedekind than that.

The fun begins when you try to determine which of the two sets a particular number belongs in.

I was trying to avoid implying that 'isolating' every number in its own partition means just two partitions, I simply thought that perhaps you were half remembering something like this.

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