Concerns about the geometry of the real number line

[Boltzmannbrain]

The real numbers cannot have a next number, but I don't understand how that can be logical.

For example, consider the segment inclusively from 1 to 2, so there are the numbers 1 and 2 at each end of the segment.  We can take off a number like 1.3 or 2 from it.  If we take the number 2 away, we are left with something like the segment 1 to the limit 2 - 1/x as x goes to infinity (or whatever it is). 

So my ultimate question is, why can we take off the end of the segment if it is something that we call 2, but we can't take off another number?  The segment only has real numbers; what makes 2 so special that it can end a segment and be removable?  

 

[Lorentz Jr]

25 minutes ago, Boltzmannbrain said:

why can we take off the end of the segment if it is something that we call 2, but we can't take off another number?  The segment only has real numbers; what makes 2 so special that it can end a segment and be removable?

Short answer: because it's math. As long as you can specify a limit of a range, you can define a range that's "open" at that end, meaning the number is excluded.

Longer answer: because the density of real numbers is infinite. There's nothing special about 2. You can use any real number to terminate a range of real numbers, and you can make the limits of the range any combination you like of open (excluded) or closed (included), but there is no "next" number immediately before or after the limits. You can also exclude a number between the limits of the range, but that breaks the range into two ranges, because there's always an infinite number of other reals between any two given reals. Not much of a gap between them, of course, but that doesn't always matter in mathematics.

Edited February 28, 2023 by Lorentz Jr

[Genady]

5 hours ago, Boltzmannbrain said:

The real numbers cannot have a next number, but I don't understand how that can be logical.

Because there is always another number between any two numbers. It is algebra:

Take any number, p. Let's assume that it has a next number. Let's call it, q. q > p.

Now take a number 0.5*(p+q). Call it, r. The number r is greater than p, r > p. It is less than q, r < q. Thus, r is between p and q. Thus, q is not next to p. Thus, p does not have a next number.

[Boltzmannbrain]

Thanks for the replies Genady and Lorentz Jr, but I am interested in one aspect of this problem.  I am interested in why the line segment should change its geometry after taking off a number from one end of it.  It seems like the number 2 in the example has a special quality geometrically speaking.  If all there is are is numbers then why is 2 an end point but not anything else?

[Genady]

49 minutes ago, Boltzmannbrain said:

Thanks for the replies Genady and Lorentz Jr, but I am interested in one aspect of this problem.  I am interested in why the line segment should change its geometry after taking off a number from one end of it.  It seems like the number 2 in the example has a special quality geometrically speaking.  If all there is are is numbers then why is 2 an end point but not anything else?

The difference is being an end point vs being an internal point. An internal point has neighbors on both sides, but an end point has a neighbor on one side only.

OTOH, you could take out any internal point, say, point 1.5. Then you get two open ends: on the left and on the right of 1.5.

Edited February 28, 2023 by Genady
Forgot your interval is [1,2]. Fixed the internal point to be in it

[studiot]

1 hour ago, Boltzmannbrain said:

Thanks for the replies Genady and Lorentz Jr, but I am interested in one aspect of this problem.  I am interested in why the line segment should change its geometry after taking off a number from one end of it.  It seems like the number 2 in the example has a special quality geometrically speaking.  If all there is are is numbers then why is 2 an end point but not anything else?

Genady and Lorenz have made some good comments but perhaps I should answer your question more formally.

The line segment does not change its geometry  - you misunderstand.

Lines, line segments and numbers are all three different things.

Lines and line segments are geometrical objects.

Numbers are not.

The real number line is a representation of all real numbers, placed into one-to-one correspondence with the real line as an assembly of points.

There is no other similar correspondence with other types of number.

We do this because it is a useful way of visualising things, especially as a lot of geometry is taught before any of the higher mathematics necessary to fully appreciate the comparison is taught.

 

The correspondence goes all the way to 'point set theory' where the elements of a set (also called points) can be one of three types.

Interior (also called accumulation points) , boundary or isolated.

Genady has already described two of these, though he called a boundary point an end point.

A set which includes its boundary points is called a closed set.

A set which does not include its boundary points is called an open set.       

An isolated point is both closed and open.

 

As regards line segments, a line segment has two boundary points and is called an interval or line interval.

An interval can be open or closed, but since it has two boundary points it may include one and not the other and we either say the interval is half open or half closed.

 

The whole real number line is open.

The line segment or interval from say zero to plus infinity or minus infinity is half open.

The interval from zero to say 2 is closed if it includes both 0 and 2 and open if it includes neither.

 

These concepts are very important when you study limits and calculus, so they have their own special notation.

Here is a diagram, showing the geometric and algebraic representation of the types interval  (0,2) the rounded brackets are the general form when the type doesn't matter.

 

 

[Luke11098]

Everything well described. Nothing more nothing less.

Edited March 2, 2023 by Luke11098

[Boltzmannbrain]

On 2/28/2023 at 9:19 AM, Genady said:

The difference is being an end point vs being an internal point. An internal point has neighbors on both sides, but an end point has a neighbor on one side only.

OTOH, you could take out any internal point, say, point 1.5. Then you get two open ends: on the left and on the right of 1.5.

I finally have time to respond.

But let's just think about this geometrically and logically for a moment.  In the case where a line segment ends at 1.5, we are able to take 1.5 from the end of a line segment.  And then we say that it no longer ends.  How does that make any sense?

[Genady]

1 minute ago, Boltzmannbrain said:

And then we say that it no longer ends

We don't say this.

We might say about the left interval that it ends at 1.5, but it does not contain the end point. It contains everything before 1.5, i.e., everything that is < 1.5, but not the point 1.5. I don't see anything not logical here. Can you point to any contradiction?

I don't see any geometrical issue at all. The length of the interval stayed the same as before.

[Boltzmannbrain]

On 2/28/2023 at 10:49 AM, studiot said:

Genady and Lorenz have made some good comments but perhaps I should answer your question more formally.

The line segment does not change its geometry  - you misunderstand.

Lines, line segments and numbers are all three different things.

Lines and line segments are geometrical objects.

Numbers are not.

 

Sorry, I meant points.

 

Quote

The real number line is a representation of all real numbers, placed into one-to-one correspondence with the real line as an assembly of points.

This is what I am having a hard time understanding.

Quote

 

The correspondence goes all the way to 'point set theory' where the elements of a set (also called points) can be one of three types.

Interior (also called accumulation points) , boundary or isolated.

Genady has already described two of these, though he called a boundary point an end point.

A set which includes its boundary points is called a closed set.

A set which does not include its boundary points is called an open set.       

An isolated point is both closed and open.

 

As regards line segments, a line segment has two boundary points and is called an interval or line interval.

An interval can be open or closed, but since it has two boundary points it may include one and not the other and we either say the interval is half open or half closed.

 

The whole real number line is open.

The line segment or interval from say zero to plus infinity or minus infinity is half open.

The interval from zero to say 2 is closed if it includes both 0 and 2 and open if it includes neither.

 

These concepts are very important when you study limits and calculus, so they have their own special notation.

Here is a diagram, showing the geometric and algebraic representation of the types interval  (0,2) the rounded brackets are the general form when the type doesn't matter.

Now my main issue is that these points that make up a line segment do not behave like points/objects in a row.  Is that maybe a what I am getting wrong?  Or can these points be seen as being in a row?

[Boltzmannbrain]

 

Quote

 

Can you point to any contradiction.

 

Yes, with what you say here, "... it ends at 1.5, but it does not contain the end point. It contains everything before 1.5 ...". 

So the points end at 1.5, but there is no end point?  If that is not a contradiction, then it is a least a very hard thing to comprehend.  

Quote

I don't see any geometrical issue at all. The length of the interval stayed the same as before.

Are you saying that some interval like [1, 1.5] is the same length as the interval [1, 1.5)?

Edited March 3, 2023 by Boltzmannbrain

[Genady]

6 hours ago, Boltzmannbrain said:

So the points end at 1.5, but there is no end point?  If that is not a contradiction, then it is a least a very hard thing to comprehend.

The phrase "the points end at 1.5" is wrong. The correct phrase is, "the interval end at 1.5". This does not contradict not containing the point 1.5.

For example, you can say that your property ends by the river, but the river is not on your property.

If you distinguish between the interval and the point, there is no contradiction.

7 hours ago, Boltzmannbrain said:

Are you saying that some interval like [1, 1.5] is the same length as the interval [1, 1.5)?

Yes, this is correct. The length is 0.5 in both cases.

[studiot]

13 hours ago, Boltzmannbrain said:

Sorry, I meant points.

The word point is much overused.

I debated with myself how to avoid it as far a possible to avoid confusion.

3 hours ago, Genady said:

If you distinguish between the interval and the point, there is no contradiction.

Totally agreed.

But I do not like the use of the word end in relation to lines as it can be imprecise.

An open interval has no end.

 

13 hours ago, Boltzmannbrain said:

This is what I am having a hard time understanding.

 

13 hours ago, Boltzmannbrain said:

Now my main issue is that these points that make up a line segment do not behave like points/objects in a row.  Is that maybe a what I am getting wrong?  Or can these points be seen as being in a row?

I am sorry you missed my main point (there is that word again, but with a different meaning this time)

So it means that my explanation was not good enough so I will try to do better.

But it is difficult to know what you know as you obviously have met the idea of open and closed intervals before since you used an alternative notation in your responses.

I carefully avoided [0,2) etc because it is easy to overlook which bracket is which and you can never be sure whether the writer meant it or not. The reversed square bracket stand out, don't you think ? Also curved brackets are used to denote sets.

Anyway you clearly understand that part.

 

My main point was that lines and numbers are not the same, although they have some properties in common, which allows one to exemplify the other when only the common properties are of concern.

But sets of numbers have lots of other properties where they cannot be represented as lines.

So mathematicians seek more general approaches.

If there is a number that is greater than any other number is the set then the set is bounded.

In fact we say it is bounded above and can say (similarly it is bounded below if there is a number less than any other number in the set.)

So the set (1, 2, 3, 4, 5, 6)  is bounded above by the numbers 6, 7, 8, 9, 10...

We call any of these an upper bound.

Also the upper bound may be an element of the set or it may not.

When the upper bound is an element we call it the maximum of the set.

 

Now there is a theorem, which I will not prove, called the least upper bound theorem.

"If any set of numbers is bounded above it has a least upper bound"

In our example 6 is the least upper bound of our set and is also the maximum.

But our set is also finite so it is easy to see this.

Finite sets means that the count of elements is finite: Infinite sets have an infinite count of elements.

The boundedness theorems apply to finite and to infinite sets, (But not the max and min)

Infinite sets can also be bounded.

The set of all the elements of the negative exponential e^-x, from x=0 to x = ∞  is bounded above by 1 and bounded below by 0, although the set is infinite becasue the count of x values is infinite.

Note the x = ∞ 'end' is never reached  or as I prefer there is no right hand end to this line.

So this line has no minimum.

The left hand end depends whether we include or exclude x = 0 in the set  (closed or open )

If we include x = 0 then the upper bound of 1 is also the maximum,

But if we exclude it then again the upper bound is never reached and the set has no maximum either.

 

 

[Genady]

29 minutes ago, studiot said:

But I do not like the use of the word end in relation to lines as it can be imprecise.

Yes, I should've replaced it from the beginning for clarity.

[Lorentz Jr]

12 hours ago, Boltzmannbrain said:

So the points end at 1.5, but there is no end point?  If that is not a contradiction, then it is a least a very hard thing to comprehend.

It's hard to comprehend because you're trying to apply the logic of finite numbers to infinities. What is an "end point"? You have a row of points, and it's the one at the end. But there may not be any "end" to the points when there's an infinite number of them.

Edited March 3, 2023 by Lorentz Jr

[joigus]

I agree with main arguments developed by @Genady, @studiot, and @Lorentz Jr. I particularly liked Studiot's summary. I would call his argument about closed and open sets --as well as those that are neither open nor closed-- a "topological approach."

A crash course in topology would include concepts such as,

Topology: Existence of an inclusion relation in a set, \( \subseteq \) --contains--, \( \subsetneq \) --does not contain. => neighbourhoods of a point.

Limit point --o accumulation point--: A point in a set that has neighbouring points also in the set that are arbitrarily close to it.

Interior of a set: All its point are limits points of the set --if I remember correctly--.

Boundary of a set: The set of all the limit points of its exterior

Closure of a set: The union of the set and ist boundary

...

etc.

With these rigorous topological definitions, when applied to the real numbers, we can prove they constitute a topological space, and, eg, the set \( \left[0,2\right]=\left\{ x\,\textrm{in}\,\mathbb{R}\,\textrm{such that}\,0\leq x\leq2\right\} \) contains its boundary --and it is, therefore, closed; while the set, eg, \( \left(0,2\right)=\left\{ x\,\textrm{in}\,\mathbb{R}\,\textrm{such that}\,0<x<2\right\} \) does not contain its boundary --and it is therefore, open.

https://en.wikipedia.org/wiki/Topological_space

 

[studiot]

On 2/28/2023 at 4:20 AM, Boltzmannbrain said:

The real numbers cannot have a next number, but I don't understand how that can be logical.

This simple scheme might help.

Which way do you want the maths to go more or less complicated?

 

Which of these real numbers is the next number after 1 ?

1.1
1.01
1.001
1.0001
1.0001
1.00001
1.000001

etc

 

Edited March 3, 2023 by studiot

[Boltzmannbrain]

5 hours ago, Genady said:

The phrase "the points end at 1.5" is wrong. The correct phrase is, "the interval end at 1.5". This does not contradict not containing the point 1.5.

I agree.  I wasn't saying that 1.5 has to be in the interval.  I was saying that since the interval of points (if I can say this) ends at 1.5, then how is there no end point.

Quote

Yes, this is correct. The length is 0.5 in both cases.

This doesn't seem right.  Imagine a line segment [0, 10].  Assume that the segment can revolve about the point 0.  At the halfway point, at 5 units, we we break the segment, keeping the number 5 on the part that revolves.  We revolve the broken segment to some degree then reattach another 5 at the end of the part that doesn't revolve.  The segment [0, 5] cannot revolve past the segment [5, 10] anymore.  The one 5 is taking up the space (even though it is 0 space) that the other 5 needs to revolve through.

[Genady]

9 minutes ago, Boltzmannbrain said:

I was saying that since the interval of points (if I can say this) ends at 1.5, then how is there no end point.

I don't think there are such concepts as "interval of points" and "interval ends at ..." in math.

 

11 minutes ago, Boltzmannbrain said:

This doesn't seem right.  Imagine a line segment [0, 10].  Assume that the segment can revolve about the point 0.  At the halfway point, at 5 units, we we break the segment, keeping the number 5 on the part that revolves.  We revolve the broken segment to some degree then reattach another 5 at the end of the part that doesn't revolve.  The segment [0, 5] cannot revolve past the segment [5, 10] anymore.  The one 5 is taking up the space (even though it is 0 space) that the other 5 needs to revolve through.

I have no idea what revolving, breaking, and reattaching segments have to do with numbers and interval lengths. The former are not mathematical concepts, AFAIK.

@Boltzmannbrain, I start to suspect that the root of confusion is here: you are talking about an actual physical segment, while the "real number line" is a mathematical concept.

The "real" in the latter does not refer to "line", i.e., it is not a "real line."

It refers to "number", i.e., they are "real numbers."

[studiot]

24 minutes ago, Genady said:

The "real" in the latter does not refer to "line", i.e., it is not a "real line."

It refers to "number", i.e., they are "real numbers."

Exactly. +1

 

 

[Boltzmannbrain]

2 hours ago, studiot said:

The word point is much overused.

I debated with myself how to avoid it as far a possible to avoid confusion.

Totally agreed.

But I do not like the use of the word end in relation to lines as it can be imprecise.

An open interval has no end.

 

 

I am sorry you missed my main point (there is that word again, but with a different meaning this time)

So it means that my explanation was not good enough so I will try to do better.

But it is difficult to know what you know as you obviously have met the idea of open and closed intervals before since you used an alternative notation in your responses.

I carefully avoided [0,2) etc because it is easy to overlook which bracket is which and you can never be sure whether the writer meant it or not. The reversed square bracket stand out, don't you think ? Also curved brackets are used to denote sets.

Anyway you clearly understand that part.

 

My main point was that lines and numbers are not the same, although they have some properties in common, which allows one to exemplify the other when only the common properties are of concern.

But sets of numbers have lots of other properties where they cannot be represented as lines.

So mathematicians seek more general approaches.

If there is a number that is greater than any other number is the set then the set is bounded.

In fact we say it is bounded above and can say (similarly it is bounded below if there is a number less than any other number in the set.)

So the set (1, 2, 3, 4, 5, 6)  is bounded above by the numbers 6, 7, 8, 9, 10...

We call any of these an upper bound.

Also the upper bound may be an element of the set or it may not.

When the upper bound is an element we call it the maximum of the set.

 

Now there is a theorem, which I will not prove, called the least upper bound theorem.

"If any set of numbers is bounded above it has a least upper bound"

In our example 6 is the least upper bound of our set and is also the maximum.

But our set is also finite so it is easy to see this.

Finite sets means that the count of elements is finite: Infinite sets have an infinite count of elements.

The boundedness theorems apply to finite and to infinite sets, (But not the max and min)

Infinite sets can also be bounded.

The set of all the elements of the negative exponential e^-x, from x=0 to x = ∞  is bounded above by 1 and bounded below by 0, although the set is infinite becasue the count of x values is infinite.

Note the x = ∞ 'end' is never reached  or as I prefer there is no right hand end to this line.

So this line has no minimum.

The left hand end depends whether we include or exclude x = 0 in the set  (closed or open )

If we include x = 0 then the upper bound of 1 is also the maximum,

But if we exclude it then again the upper bound is never reached and the set has no maximum either.

 

 

Thanks a lot for this.  I did take an advanced calculus course that covered maximums, minimums, boundaries, etc, but I have long forgotten most of it.

Using "bounded" instead of "ends" does answer some of my concerns, but I still fell like there is more for me to learn about this part of mathematics.  In other words, I still have ideas that are not answer by what I know up to this point about this topic.  I will look at my old notes.

 

 

2 hours ago, Lorentz Jr said:

It's hard to comprehend because you're trying to apply the logic of finite numbers to infinities. What is an "end point"? You have a row of points, and it's the one at the end. But there may not be any "end" to the points when there's an infinite number of them.

Yes, but what keeps me curious is when infinite becomes finite.  For example, an infinite number of numbers/points has a finite distance. 

[studiot]

3 minutes ago, Boltzmannbrain said:

Thanks a lot for this.  I did take an advanced calculus course that covered maximums, minimums, boundaries, etc, but I have long forgotten most of it.

Using "bounded" instead of "ends" does answer some of my concerns, but I still fell like there is more for me to learn about this part of mathematics.  In other words, I still have ideas that are not answer by what I know up to this point about this topic.  I will look at my old notes.

 

 

Sometimes it is just a question of getting the right words as using popular general ones in their non scientific sense can lead to misunderstandings.

🙂

[Boltzmannbrain]

1 hour ago, joigus said:

I agree with main arguments developed by @Genady, @studiot, and @Lorentz Jr. I particularly liked Studiot's summary. I would call his argument about closed and open sets --as well as those that are neither open nor closed-- a "topological approach."

A crash course in topology would include concepts such as,

Topology: Existence of an inclusion relation in a set, ⊆ --contains--, ⊊ --does not contain. => neighbourhoods of a point.

Limit point --o accumulation point--: A point in a set that has neighbouring points also in the set that are arbitrarily close to it.

Interior of a set: All its point are limits points of the set --if I remember correctly--.

Boundary of a set: The set of all the limit points of its exterior

Closure of a set: The union of the set and ist boundary

...

etc.

With these rigorous topological definitions, when applied to the real numbers, we can prove they constitute a topological space, and, eg, the set [0,2]={xinRsuch that0≤x≤2} contains its boundary --and it is, therefore, closed; while the set, eg, (0,2)={xinRsuch that0<x<2}

Ok, this is helpful.  I will look at some of these concepts more carefully.  Thanks.

[Lorentz Jr]

29 minutes ago, Boltzmannbrain said:

Yes, but what keeps me curious is when infinite becomes finite.  For example, an infinite number of numbers/points has a finite distance.

That doesn't mean it's finite. The density of points is infinity divided by a finite number, which is also infinite.

If the length of the closed interval is L, then you can think of the length of the open interval as the limit of L - dx as dx approaches zero. The difference is a finite number divided by infinity.

[Boltzmannbrain]

2 hours ago, studiot said:

This simple scheme might help.

Which way do you want the maths to go more or less complicated?

 

Which of these real numbers is the next number after 1 ?

1.1
1.01
1.001
1.0001
1.0001
1.00001
1.000001

etc

 

This example has always been my intuition when thinking about a "next real number".  However, this only shows that this particular method to find a next number does not work. 

What if there were an infinite number of 0's after the decimal.  If we can do this, (although I can't imagine there is a function that would allow this) then we could theoretically get to a smallest real number, specifically, 0.000 ... 0001.  And it would be defined to be the smallest like the natural numbers define 1 (at least in the more common definition of the naturals) to be its smallest.  So you couldn't divide 0.000 ... 0001 by 2 to get a real number, just like you can't divide 1 by 2 to get a natural number.