Concerns about the geometry of the real number line

[studiot]

19 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 here is that, after we go beyond the integers,  we are never looking for a 'next' number because we never start with just one number.  We need to employ two numbers.

It works like this.

All number systems inherit properties from simpler systems from which they are built.

We cannot build the integers without the natural numbers.

We cannot build the rational numbers without the integers.

I will take these as read and proceed to the rational numbers.

We define a rational number as the ratio of two integers. It has to be integers since we are defining 'fractions' so we cannot employ fractions in our definition or it would be a circular definition.

 

We define one rational number a/b as being greater than another , c / d,          where a,b,c,d are all integers

[math]\frac{a}{b} > \frac{c}{d}\quad if\quad ad > bc[/math]

We can use this to find a third rational number between any two rational numbers as

[math]\frac{a}{b} > \frac{{a + mc}}{{b + md}} > \frac{c}{d}\quad if\quad b,d,m\;are\;positive\;{\mathop{\rm int}} egers[/math]

 

I suggest you practice some algebra by convincing yourself my formulae are good.

We can repeat this process indefinitely, placing a fourth rational number between our first and third or second and third rational numbers.

This leads to the conclusion that between any two rationals there is another and by extension infinitely many other rationals.

This density property amounts to saying that, for the rationals, there is no 'next' number to any rational since given any proposed next rational we can place another closer, by this process.

So let us move on to the set (I learned class) of all rational numbers and divide it anywhere into two subsets.

This is actually where a number line comes in handy since we can call the subset to the left of our dividing line L (with typical element l) and the subset to the right R( with typical element r).

Because we have divided the line to the left and right, any l < any r.

 

Now for the clever bit.

If we pick one rational from each of L and R, say l and r then we can examine the infinity of rationals between l and r

according to our formula above.

Each and every one of these must individually belong to either L or to R, but not to both.

There can be no rational numbers belonging to both L and R.

This means that we have an infinite sequence of increasing rational numbers greater than l in L

And an infinite decreasing sequence of rational numbers less than r in R

So L has no greatest element
And R has no least element

So we have created two open sets and sequences a la Genady.

This density property is inherited by the reals from the rationals, as noted at the beginning of this post.

As also noted the fun starts when we try to incorporate  irrational numbers into the scheme as the next post will show, with the story of root2  - the first irrational to be discovered.
Again this is where geometry helps explain things.

 

 

Edited March 8, 2023 by studiot

[Boltzmannbrain]

10 hours ago, wtf said:

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+1−x2 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.

 

But what if we do this process an infinite number of times?

Quote

 

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.

 

I am trying.  Except when when I think about the reals as continuous (which they are) rather than granular, the following issue pops up in my brain, "if the reals are a continuum (which they are), then the boundary must be attached to another number since there isn't anything else it could be attached to.  I will explain more below.

Quote

 

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.

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

 

Quote

 

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. 

 

This is what I struggling to understand.  The set meets 1 in a continuous and complete manner.  Ordered objects in an increasing fashion connect to 1.  Only one object can be larger than the rest but smaller than 1, just like the number 1 is to the rest of the set.  But somehow this object that I am looking for gets lost in the fuzz. 

13 hours ago, Genady said:

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.

That is a perfect analogy.  Removing 1 from [0, 1] is just like moving the top layer of water as well as its surface.  That is why this does not make sense to me geometrically.

The numbers/objects are like the H2O molecules.  Remove the top layer and there should be a next layer of molecules, but there isn't.

12 hours ago, studiot said:

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 ?

 

I only want to understand this using "mainstream mathematics" which I think is called Zermelo-Fraenko set theory.  

Quote

 

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.

 

I know that aleph null is not in the set of naturals since it is not a natural number.  

Quote

 

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.

 

I see you have posted about this, so I will read your next post.

[Genady]

21 minutes ago, Boltzmannbrain said:

14 hours ago, Genady said:

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.

That is a perfect analogy.  Removing 1 from [0, 1] is just like moving the top layer of water as well as its surface.  That is why this does not make sense to me geometrically.

The numbers/objects are like the H2O molecules.  Remove the top layer and there should be a next layer of molecules, but there isn't.

I am sorry you don't understand my analogy.

[Boltzmannbrain]

5 hours ago, studiot said:

The point here is that, after we go beyond the integers,  we are never looking for a 'next' number because we never start with just one number.  We need to employ two numbers.

It works like this.

All number systems inherit properties from simpler systems from which they are built.

We cannot build the integers without the natural numbers.

We cannot build the rational numbers without the integers.

I will take these as read and proceed to the rational numbers.

We define a rational number as the ratio of two integers. It has to be integers since we are defining 'fractions' so we cannot employ fractions in our definition or it would be a circular definition.

 

We define one rational number a/b as being greater than another , c / d,          where a,b,c,d are all integers

ab>cdifad>bc

We can use this to find a third rational number between any two rational numbers as

ab>a+mcb+md>cdifb,d,marepositiveintegers

 

I suggest you practice some algebra by convincing yourself my formulae are good.

We can repeat this process indefinitely, placing a fourth rational number between our first and third or second and third rational numbers.

This leads to the conclusion that between any two rationals there is another and by extension infinitely many other rationals.

This density property amounts to saying that, for the rationals, there is no 'next' number to any rational since given any proposed next rational we can place another closer, by this process.

So let us move on to the set (I learned class) of all rational numbers and divide it anywhere into two subsets.

This is actually where a number line comes in handy since we can call the subset to the left of our dividing line L (with typical element l) and the subset to the right R( with typical element r).

Because we have divided the line to the left and right, any l < any r.

 

Now for the clever bit.

If we pick one rational from each of L and R, say l and r then we can examine the infinity of rationals between l and r

according to our formula above.

Each and every one of these must individually belong to either L or to R, but not to both.

There can be no rational numbers belonging to both L and R.

This means that we have an infinite sequence of increasing rational numbers greater than l in L

And an infinite decreasing sequence of rational numbers less than r in R

So L has no greatest element
And R has no least element

So we have created two open sets and sequences a la Genady.

This density property is inherited by the reals from the rationals, as noted at the beginning of this post.

As also noted the fun starts when we try to incorporate  irrational numbers into the scheme as the next post will show, with the story of root2  - the first irrational to be discovered.
Again this is where geometry helps explain things.

 

 

I understand.  This kind of argument seems a lot like the other arguments where there is always a number between two other numbers.  This definitely works for a finite number of trials.  But what about an infinite number trials.  Don't we exhaust all divisions between the two numbers?

[wtf]

7 hours ago, Boltzmannbrain said:

But what if we do this process an infinite number of times?

Say we start at x = 1/2. Then repeating the "half the distance" idea gives 3/4, 7/8, 15/16, 31/32, ... We get an infinite sequence of points, each one a little closer to 1, but none of them is the largest in [0,1).

 

7 hours ago, Boltzmannbrain said:

I am trying.  Except when when I think about the reals as continuous (which they are) rather than granular, the following issue pops up in my brain, "if the reals are a continuum (which they are), then the boundary must be attached to another number since there isn't anything else it could be attached to.  I will explain more below.

This is vague. Numbers aren't "attached to" each other as if by velcro for example. Didn't the two-dimensional example of the boundary of a circle help? 

Sometimes geometric intuition is helpful. Sometimes it's confusing. What is the formal definition of [0,1)? It's 

[math]\{x \in \mathbb R : 0 \leq x < 1\}[/math]

That denotes the set of all real numbers greater than or equal to 0, and strictly less than 1. There's no "attachment" in the math. Just a set of real numbers. 

By the way this is set theory notation, since you mentioned Zermelo-Fraenkel. The curly brackets denote a set. The colon is read, "such that." The boldface [math]\mathbb R[/math] stands for the real numbers. So the notation literally denotes "The set of all real numbers x such that" etc.

 

7 hours ago, Boltzmannbrain said:

That is a perfect analogy.  Removing 1 from [0, 1] is just like moving the top layer of water as well as its surface.  That is why this does not make sense to me geometrically.

The numbers/objects are like the H2O molecules.  Remove the top layer and there should be a next layer of molecules, but there isn't.

This analogy doesn't work for me. For one thing, water isn't the real numbers. Water is made of discrete molecules and exists in the physical world; while the real numbers are a mathematical abstraction that, as you note, form a continuum.

But even on its own terms, the water analogy doesn't work. If you zoom in to the "surface" of a body of water, you find a cloud of molecules that aren't part of the body of water, but are in the process of evaporating into the air. In fact the surface of a body of water very nicely models an open set with no boundary at all. The surface is fuzzy, with molecules constantly jumping around and some of them escaping into the surrounding air.

 

 

7 hours ago, Genady said:

I am sorry you don't understand my analogy.

I don't understand it either, as shown by my example of zooming in to the "surface" to see a cloud of molecules leaving the water and evaporating into the air. There is no well-defined surface of a body of water. Now perhaps you are thinking of surface tension, but still, the analogy is strained. If you zoom in, you see molecules bouncing around. You can never point to any one collection of molecules and say, "That's the surface." Before you're done speaking, some of those molecules have evaporated into the air. 

Edited March 9, 2023 by wtf

[Genady]

1 hour ago, wtf said:

8 hours ago, Genady said:

I am sorry you don't understand my analogy.

I don't understand it either, as shown by my example of zooming in to the "surface" to see a cloud of molecules leaving the water and evaporating into the air. There is no well-defined surface of a body of water. Now perhaps you are thinking of surface tension, but still, the analogy is strained. If you zoom in, you see molecules bouncing around. You can never point to any one collection of molecules and say, "That's the surface." Before you're done speaking, some of those molecules have evaporated into the air. 

I didn't mention molecules or atoms and I didn't mean to relate to a physical structure of water surface in any way. I could talk about a table surface or a mirror surface as well. The point was purely geometrical: to consider a relation between two-dimensional surface and three-dimensional layer. Thought it might be easier to visualize. But, if analogy doesn't work, ignore it.

[joigus]

Numbers (real numbers) are not atoms, they're not beads on a string, they're not pieces of code on a DNA strand, they're not characters on an alphanumeric string. They're abstractions. That's, I think, at the root of why you're finding it so hard to wrap your head around the ideas that define them. Same reason why people I will describe in anecdote a1) at the end of this post, completely missed the point too. So you're not alone.

Axiom of completeness:

(1) Semi-intuitive notion: Real numbers have no gaps or holes

(2) Rigorous notion: Every nonempty subset \( X \) of \( \mathbb{R} \) that is bounded above has a least upper bound. That is, \( \sup X \) exists and is a real number.

Consider this sequence --not made up of atoms, nor made up of beads, nor made of little LED lights; made up of numbers--,

\[ X=\left\{ x\in\mathbb{Q}/x=1-\frac{1}{n},n\in\mathbb{N}\right\} \]

In other words,

\[ X=\left\{ 0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},\frac{6}{7},\frac{7}{8},\cdots\right\} \]

Now consider the following facts:

A) Every number in \( X \) is less than 1.

B) For every \( r \) such that every number in \( X \) is less than \( r \), \( r\geq1 \). That means that 1 is not only an upper bound to \( X \); it is, actually, the best upper bound we can find. That is, 1 is the least upper bound.

Challenge: Find a number in \( X  \) that gives you exactly 1.

If you get to understand how you will fail to find that number, say \( x_{n}=1-\frac{1}{n} \), you will be a step closer to understanding this logical frustration.

Let me finish with a couple of anecdotes:

a1) I was once sitting at a Calculus class and the professor told us about the axiom of completenes in the form, "every monotonically increasing sequence in the real numbers possesses a least upper bound."

A couple of students in front of me started giggling and went something like, "Doh! Why of course." Needless to say, they'd completely missed the point.

a2) Our Electricity and Magnetism professor said. "Imagine an R, RC, or LRC circuit with an amperimeter. You measure the current and it gives 1.3 Amperes. What kind of number is that? Is it real, rational, imaginary? After a prolongued murmur somebody uttered: It's a real number.

The professor said: "No, it's neither one of them. It's a measured number." 

Don't let the adjective 'real' deceive you. They're real alright. In some sense.

[Genady]

13 minutes ago, joigus said:

a1)

a2)

a3) One of my professors once said something to this effect (I don't remember exact words): Imaginary numbers are in no ways less real than real numbers.

[joigus]

3 minutes ago, Genady said:

a3) One of my professors once said something to this effect (I don't remember exact words): Imaginary numbers are in no ways less real than real numbers.

Exactly. I agree 100% with your professor. I have no qualms about considering the refraction index as a complex number, and its real and imaginary parts as real numbers. What I try not to forget, ever, is that there is a corpus of theory undelying this idea. It could just be approximately right. A real number is a convenient tool that gives you room to accomodate infinite precision. What's not to like about this wonderful tool?

[Genady]

4 minutes ago, joigus said:

Exactly. I agree 100% with your professor. I have no qualms about considering the refraction index as a complex number, and its real and imaginary parts as real numbers. What I try not to forget, ever, is that there is a corpus of theory undelying this idea. It could just be approximately right. A real number is a convenient tool that gives you room to accomodate infinite precision. What's not to like about this wonderful tool?

He was a good professor.

It was not a topic of that class, but I'd add that:

Rational numbers, integers, or natural numbers are in no way more real than real or imaginary numbers.

[studiot]

33 minutes ago, joigus said:

Numbers (real numbers) are not atoms, they're not beads on a string, they're not pieces of code on a DNA strand, they're not characters on an alphanumeric string.

etc

Nice. +1

11 hours ago, Boltzmannbrain said:

I know that aleph null is not in the set of naturals since it is not a natural number.  

Good

 

Infinity is not a number. So it cannot be a member of either the set of reals or the set of naturals or the set of rationals or the set of integers.

So why do you claim infinity is a number ?

11 hours ago, Boltzmannbrain said:

But what if we do this process an infinite number of times?

 

None of the elements of set of reals or the set of naturals or the set of rationals or the set of integers are infinite.

You are confusing the sets themselves which are infinite, whith the elements of those sets

The elements (ie the numbers themselves) are all finite, without exception.

 

You misuderstand what is meant by infinity.

 

So it is as pointless to want to use ZF theory as saying "I have never used a chainsaw, but I am going to take one, climb a 30 foot tree and start cutting".

11 hours ago, Boltzmannbrain said:

I only want to understand this using "mainstream mathematics" which I think is called Zermelo-Fraenko set theory.

I was so very disappointed with this response that I considered your 'bail' option as I am no longer sure whether you are only trying to find fault here or actually trying to understan.

Understanding takes effort on the part of the student.

Yet you have uttered not a single word about inheriting key properties.

Have you tried the simple proof I suggested ?

You suddenly mention continuums.  These are not like continuums in 'continuums' in physics or 'continuum mechanics'.

The 'Continuum Hypothesis'  is suprising in what is actually hypothesised,

Just as the ZF axioms are suprising in their names and statements.

Have you ever worked with any ?

 

[Mc2509]

On 3/7/2023 at 11:57 PM, Boltzmannbrain said:

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

 

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.

 

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

That makes sense.

Interesting!

Sure that sounds useful and interesting.

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.

A real number line is a continuous line that extends infinitely in both directions. Each point on the line represents a real number. The real numbers on the line are arranged in sequential order. Any two real numbers on the line can be compared, and we can determine which number is greater or smaller than the other. The line can be divided into segments.

Suppose we take a segment on the real number line, let us say between two integers, 1 and 2. The segment is finite but infinite in the number of real numbers it contains. There are countless real numbers between 1 and 2. However, there is no next number on this segment. By this, we mean that if you pick any number between 1 and 2, you can always find another number between them. There is no limit or endpoint to the number of real numbers between the two integers. Thus, there can never be a next number in this segment.

The concept of the next number is not applicable to the real number line. The set of real numbers is complete, meaning that there is no need for any new number to fill in any gaps or provide solutions to any problems. Any two real numbers on the line are separated by an infinite number of other real numbers. This implies that there exists no next number or any number missing from the real number line. Each number on the line is unique and independent.

Furthermore, the real number line is dense, which means that every point on the line can be approached arbitrarily close by a sequence of real numbers. This property further emphasizes that there is no next number on the real number line. For any point on the line, we can approach it arbitrarily close by finding real numbers that are infinitely close to the point. Since there exists no smallest positive number on the real number line, there is no next number.

there is an end to the real number line segment or not?

At first glance, it may seem that the real number line goes on forever without any end. This idea aligns with the concept of infinity, which is an unbounded quantity or magnitude that extends indefinitely without a limit or boundary. Mathematically, we can represent infinity by the symbol ∞, which denotes an infinitely large or small value that cannot be expressed or reached in the usual sense. In this sense, the real number line appears to be infinite both to the left and right of zero, with no end in sight.

However, upon further analysis, we realize that there are limits to the real number line, albeit they may not be intuitive or straightforward. For instance, we can define bounds or intervals on the real number line that contain only a finite or countable number of real numbers. For example, we can define the interval [0,1], which contains all real numbers between zero and one, including both endpoints. In this case, the interval [0,1] is bounded, meaning it has an upper and lower limit, which are 1 and 0, respectively.

Furthermore, there are situations where the real number line is incomplete or non-existent in certain spots. For instance, we can consider the imaginary or complex number system, which extends the real number line by introducing the imaginary unit i, such that i^2=-1. The complex number system includes both real and imaginary numbers, and it can be represented as a two-dimensional plane with a horizontal axis for the real part and a vertical axis for the imaginary part. However, there are points on the complex plane where the real part is zero, and only the imaginary part exists. Such points are called purely imaginary or vertical lines and are not part of the real number line. Therefore, the real number line is incomplete or does not exist in such cases.

Another situation where the real number line segment is incomplete is in the case of limits or approaches to infinity. For example, we can consider the function f(x)=1/x, which approaches zero as x approaches infinity. In this case, the real number line seems to have an end or limit at zero, but we can still consider values greater than zero by taking the limit as x approaches infinity, which yields a value of zero. Therefore, the real number line segment has a limit or end but extends infinitely beyond it.

[studiot]

7 hours ago, Mc2509 said:

Suppose we take a segment on the real number line, let us say between two integers, 1 and 2. The segment is finite but infinite in the number of real numbers it contains. There are countless real numbers between 1 and 2.

 

7 hours ago, Mc2509 said:

For instance, we can define bounds or intervals on the real number line that contain only a finite or countable number of real numbers. For example, we can define the interval [0,1], which contains all real numbers between zero and one, including both endpoints. In this case, the interval [0,1] is bounded, meaning it has an upper and lower limit, which are 1 and 0, respectively.

 

So which is it - countable or uncountable ?

[Boltzmannbrain]

On 3/9/2023 at 1:04 AM, wtf said:

Say we start at x = 1/2. Then repeating the "half the distance" idea gives 3/4, 7/8, 15/16, 31/32, ... We get an infinite sequence of points, each one a little closer to 1, but none of them is the largest in [0,1).

 

Ok, this answers that issue.  

 

Quote

This is vague. Numbers aren't "attached to" each other as if by velcro for example.

Didn't the two-dimensional example of the boundary of a circle help? 

 

I was thinking that a continuum meant that the points/numbers had to be connected or attached to each other, but now after reading about continuity and the continuum I don't think that is necessary.

 

Quote

 

This analogy doesn't work for me. For one thing, water isn't the real numbers. Water is made of discrete molecules and exists in the physical world; while the real numbers are a mathematical abstraction that, as you note, form a continuum.

But even on its own terms, the water analogy doesn't work. If you zoom in to the "surface" of a body of water, you find a cloud of molecules that aren't part of the body of water, but are in the process of evaporating into the air. In fact the surface of a body of water very nicely models an open set with no boundary at all. The surface is fuzzy, with molecules constantly jumping around and some of them escaping into the surrounding air.

 

The general idea is that the interval [0, 1] is very different after taking just one number away to make it [0, 1). 

Say you strip a top layer off an onion with infinite layers of infinite density, shouldn't there be another layer to take off?

 

 

Edited March 10, 2023 by Boltzmannbrain

[Genady]

19 minutes ago, Boltzmannbrain said:

The general idea is that the interval [0, 1] is very different after taking just one number away to make it [0, 1). 

Say you strip a top layer off an onion with infinite layers of infinite density, shouldn't there be another layer to take off?

Yes, they - [0.1] and [0,1) - are very different.

Another layer to take off has to have a finite depth.

Edited March 10, 2023 by Genady

[Boltzmannbrain]

On 3/9/2023 at 4:19 AM, joigus said:

Numbers (real numbers) are not atoms, they're not beads on a string, they're not pieces of code on a DNA strand, they're not characters on an alphanumeric string. They're abstractions.

Yes, I definitely agree with this.  And you bring up an important point: what are their abstract properties as they exist on the number line, a segment, in a space, etc?  Thinking about this might help me come to a conclusion. 

From what I understand, a number seems to be a location on a number line or even a segment.  Now do they have to "take up" a zero dimensional point or can they just exist there with no point?   

From what I have been reading about the linear continuum, it seems as though the point and the number are tied together.  I cautiously come to that conclusion since a linear continuum is ordered and that each point represents a unique number and the converse.  

On 3/9/2023 at 4:19 AM, joigus said:

 

 

Now consider the following facts:

A) Every number in X is less than 1.

B) For every r such that every number in X is less than r , r≥1 . That means that 1 is not only an upper bound to X ; it is, actually, the best upper bound we can find. That is, 1 is the least upper bound.

Challenge: Find a number in X that gives you exactly 1.

If you get to understand how you will fail to find that number, say xn=1−1n , you will be a step closer to understanding this logical frustration.

Let me finish with a couple of anecdotes:

a1) I was once sitting at a Calculus class and the professor told us about the axiom of completenes in the form, "every monotonically increasing sequence in the real numbers possesses a least upper bound."

A couple of students in front of me started giggling and went something like, "Doh! Why of course." Needless to say, they'd completely missed the point.

 

I actually did learn about this in university too.  It was a more fundamental calculus course than what is needed for the sciences.  But unfortunately I only took the one semester.  Thanks though these concepts are great reminders for this topic.

[joigus]

27 minutes ago, Boltzmannbrain said:

From what I understand, a number seems to be a location on a number line or even a segment.  Now do they have to "take up" a zero dimensional point or can they just exist there with no point?   

Not necessarily. The set of integers modulo 7 are numbers. And very useful ones at that. And there's nothing in their nature that even remotely suggests a line. They are quite independent from the concept of a point.

Some numbers can be assimilated to topological and geometric concepts. But they don't have to. A line has no zero point. What is the origin of a geometric line?

Geometry is one thing

Topology is another

And algebra, yet another

Mathematicians love to play with these things. I'm sure @studiot and @wtf can tell you volumes about it.

They go like: Can I drop this property and still get something interesting?

[Boltzmannbrain]

On 3/9/2023 at 5:21 AM, studiot said:

 

Infinity is not a number. So it cannot be a member of either the set of reals or the set of naturals or the set of rationals or the set of integers.

So why do you claim infinity is a number ?

 

 

I actually remember the whole context of what he said.  He said that infinity is not a number, then said, well, it can be a number, but for the purposes of this course, it is a direction.  We did not get to learn why he said it was also a number.

 

Quote

 

None of the elements of set of reals or the set of naturals or the set of rationals or the set of integers are infinite.

You are confusing the sets themselves which are infinite, whith the elements of those sets

The elements (ie the numbers themselves) are all finite, without exception.

 

Yes, I agree.  This is true by definition.  I don't understand what made you have to say this.

I responded to WTF's comment " He noted that if p and q are real numbers then (p + q)/2 is a real number strictly between them." 

with this,

"But what if we do this process an infinite number of times?"

 

Quote

I was so very disappointed with this response that I considered your 'bail' option as I am no longer sure whether you are only trying to find fault here or actually trying to understan.

I thought you would have been satisfied with my answer.  Wikipedia says, 

"Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.". 

I want to use standard analysis, not nonstandard analysis for topic (if we can).  

Quote

 

Understanding takes effort on the part of the student.

Yet you have uttered not a single word about inheriting key properties.

Have you tried the simple proof I suggested ?

You suddenly mention continuums.  These are not like continuums in 'continuums' in physics or 'continuum mechanics'.

The 'Continuum Hypothesis'  is suprising in what is actually hypothesised,

Just as the ZF axioms are suprising in their names and statements.

Have you ever worked with any ?

 

I have just read through them, and I definitely have not worked with them.

[wtf]

2 hours ago, Boltzmannbrain said:

Ok, this answers that issue.  

This was in response to your question about what happens if we continually halve the distance to the end of [0, 1). But if you accept that, why are you still confused about halving the distance between p and q?

 

 

2 hours ago, Boltzmannbrain said:

I was thinking that a continuum meant that the points/numbers had to be connected or attached to each other, but now after reading about continuity and the continuum I don't think that is necessary.

 

In set theory, the continuum is just another name for the set of real numbers. In topology, a continuum is "a nonempty compact connected metric space." Either way, it's not entirely helpful to try to reason from the everyday or philosophical meaning of the word. 

https://en.wikipedia.org/wiki/Continuum_(topology)

By the way, the set of real numbers is not compact. So according to Wikipedia, the real numbers are not a continuum. That's contrary to pretty much everybody. You have to take Wikipedia with a large grain of sodium chloride.

 

2 hours ago, Boltzmannbrain said:

The general idea is that the interval [0, 1] is very different after taking just one number away to make it [0, 1). 

In some ways yes, in some ways no. In terms of cardinality, they are exactly the same. In terms of length, they are exactly the same. In terms of topology, [0,1] is a compact set, which has many important properties that [0,1) lacks. For example any continuous function on a compact set must necessarily attain its maximum and minimum. This is not true of [0,1). So when you remove the end point some things don't change and other things do. Topologically, removing that one endpoint makes a huge difference.

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

 

2 hours ago, Boltzmannbrain said:

Say you strip a top layer off an onion with infinite layers of infinite density, shouldn't there be another layer to take off?

Is this a physical onion, or an imaginary onion in your mind? If it's a physical onion, it does not have infinitely many layers. If it's imaginary, I don't know what you're imagining.

But explain me this. Did you understand my earlier demonstration that no matter what number you claim is the largest in [0,1), it turns out that it is NOT the largest. Did you understand that? If so, why are you still tossing out imaginary onions? And if not, which part is unclear or unconvincing?

One perfectly sensible response on your part would be, "Oh, I see. There can not logically be any largest element of [0,1). I shall adjust my intuitions accordingly." That's the purpose of the exercise, to sharpen and correct our intuitions. Not to talk about hypothetical imaginary onions after you've been shown a proof. If you dispute the proof, let's focus on that. Having seen the proof, why are you still insisting on an intuition that is falsified by the proof?

Let's nail down the understanding of the proof that there is no largest number in [0,1). Once we do that, then it will be clear that all intuitions to the contrary are inaccurate.

 

1 hour ago, Boltzmannbrain said:

I actually remember the whole context of what he said.  He said that infinity is not a number, then said, well, it can be a number, but for the purposes of this course, it is a direction.  We did not get to learn why he said it was also a number.

 

What he meant is that in higher set theory, we can study transfinite numbers, like the transfinite cardinals [math]\aleph_0, \aleph_1[/math], etc., and the transfinite ordinals ω,ω+1, , and so forth. These are far outside of the scope of our discussion, but that's what your professor was referring to.

 

1 hour ago, Boltzmannbrain said:

I responded to WTF's comment " He noted that if p and q are real numbers then (p + q)/2 is a real number strictly between them." 

with this,

"But what if we do this process an infinite number of times?"

That was @Genady's comment on page one of this thread, credit where due.

But if you understand that we can continually split the difference 1/2, 3/4, 7/8, etc., why are you still unclear about this? It's the same idea. 

 

 

1 hour ago, Boltzmannbrain said:

I thought you would have been satisfied with my answer.  Wikipedia says, 

"Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics.". 

I want to use standard analysis, not nonstandard analysis for topic (if we can).  

Well FWIW nonstandard analysis is also a part of ZFC. But we are talking about the standard reals and need not go any further than that. 

Edited March 10, 2023 by wtf

[studiot]

17 minutes ago, wtf said:

In some ways yes, in some ways no. In terms of cardinality, they are exactly the same. In terms of length, they are exactly the same. In terms of topology, [0,1] is a compact set, which has many important properties that [0,1) lacks. For example any continuous function on a compact set must necessarily attain its maximum and minimum. This is not true of [0,1). So when you remove the end point some things don't change and other things do. Topologically, removing that one endpoint makes a huge difference.

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

This is a beautifully crafted paragraph that should be studied very carefully as it contains lots of useful and important information.

+1

[joigus]

2 minutes ago, studiot said:

This is a beautifully crafted paragraph that should be studied very carefully as it contains lots of useful and important information.

+1

Agreed.

[Boltzmannbrain]

On 3/10/2023 at 1:49 PM, wtf said:

This was in response to your question about what happens if we continually halve the distance to the end of [0, 1). But if you accept that, why are you still confused about halving the distance between p and q?

I was thinking that we could only use finite numbers to "inch" closer and closer to 1.  If that is the case then ok, but I need to know the exact logic about why we can't use transfinite or infinite numbers.  If we can use transfinite or infinite numbers, then it would seem that there are decreasing subsets all the way to the end of [1, 0) with it ending with a subset of 0 numbers.  That final subset of 0 numbers between two numbers is what I am looking for.

 

Quote

In some ways yes, in some ways no. In terms of cardinality, they are exactly the same. In terms of length, they are exactly the same. In terms of topology, [0,1] is a compact set, which has many important properties that [0,1) lacks. For example any continuous function on a compact set must necessarily attain its maximum and minimum. This is not true of [0,1). So when you remove the end point some things don't change and other things do. Topologically, removing that one endpoint makes a huge difference.

I agree with everything you said except for the "exactly the same" length part.  This came up before in this thread.  I said something like this.  

Imagine a line segment [0, 6]  (please bare with me as I don't know how to use the proper notation for a line segment on this forum).  Assume that the segment can pivot about the point 0.  At the point 5 we break the segment, keeping the number 5 on the part that pivots, and leaving the part (5, 6].  We pivot the broken segment [0, 5] to some degree then reattach another 5 at the end of the part that doesn't pivot.  The segment [0, 5] cannot revolve past the segment new [5, 6] anymore.  The one 5 is taking up the space (even though it is 0 space) that the other 5 needs to pass through.

It would seem geometrically that there is more length on [0, 5] than [0, 5)

Quote

Is this a physical onion, or an imaginary onion in your mind? If it's a physical onion, it does not have infinitely many layers. If it's imaginary, I don't know what you're imagining.

It would just be an imaginary onion that is infinitely dense with an infinite number of layers.  You peel one layer back, and you would expect a next layer, right?

Quote

But explain me this. Did you understand my earlier demonstration that no matter what number you claim is the largest in [0,1), it turns out that it is NOT the largest. Did you understand that? If so, why are you still tossing out imaginary onions? And if not, which part is unclear or unconvincing?

Yes, I have always understood that.  My counter arguments that pop up in my head leave me unconvinced and also curious as to what it is that I still do not understand about this topic.

 

Quote

One perfectly sensible response on your part would be, "Oh, I see. There can not logically be any largest element of [0,1). I shall adjust my intuitions accordingly." That's the purpose of the exercise, to sharpen and correct our intuitions. Not to talk about hypothetical imaginary onions after you've been shown a proof. If you dispute the proof, let's focus on that. Having seen the proof, why are you still insisting on an intuition that is falsified by the proof?

And I appreciate this a lot.  I have learnt a lot since starting this thread.

This is just a long shot, but what if we discover something?  That would be great for all of us since we have all worked on this together.

Quote

Let's nail down the understanding of the proof that there is no largest number in [0,1). Once we do that, then it will be clear that all intuitions to the contrary are inaccurate.

Are you sure they are proofs?  If so, what kind of proofs are they? Mathematical proof - Wikipedia  I am quite familiar with most of them, but not all.  

 

Edited March 11, 2023 by Boltzmannbrain

[wtf]

3 hours ago, Boltzmannbrain said:

I was thinking that we could only use finite numbers to "inch" closer and closer to 1.  If that is the case then ok, but I need to know the exact logic about why we can't use transfinite or infinite numbers.

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?
 

 

3 hours ago, Boltzmannbrain said:

If we can use transfinite or infinite numbers, then it would seem that there are decreasing subsets all the way to the end of [1, 0) with it ending with a subset of 0 numbers.  That final subset of 0 numbers between two numbers is what I am looking for.

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.

 

3 hours ago, Boltzmannbrain said:

I agree with everything you said except for the "exactly the same" length part.  This came up before in this thread. 

Ah. Well, the length of a single point is zero. The length of [0,1] is 1, and so is the length of [0,1). The addition or deletion of a single point makes no difference when we're calculating the length of an interval. 

 

 

3 hours ago, Boltzmannbrain said:

Imagine a line segment [0, 6]  (please bare with me

I'll keep my clothes on if it's all the same to you, thanks.

 

3 hours ago, Boltzmannbrain said:

Assume that the segment can pivot about the point 0.  At the point 5 we break the segment, keeping the number 5 on the part that pivots, and leaving the part (5, 6].  We pivot the broken segment [0, 5] to some degree then reattach another 5 at the end of the part that doesn't pivot.  The segment [0, 5] cannot revolve past the segment new [5, 6] anymore.  The one 5 is taking up the space (even though it is 0 space) that the other 5 needs to pass through.

It would seem geometrically that there is more length on [0, 5] than [0, 5)

The length of a point is zero, so adding or deleting a point can not make any difference in the length of a line segment.

I think what you are saying is that the intervals [0,5] and [5,6] both contain the number 5, and that is correct. I don't see how that helps you to find a largest number in [0,1). So the two intervals would "bang into each other" at the point 5. 

You are correct that 5 is an element of both intervals. But as a point, the number 5 has length 0. The two facts are both true. 5 is a point on the real number line and it has length 0. 

So yes, two points of zero length can still bang into each other, if you want to put it that way. Remember, Newton showed that you can reduce gravitational calculations to "point masses." So if it helps, you can think of them that way. They are points with zero dimensions, zero length, and zero volume, but they still pack a punch. I'm not saying that's any kind of mathematical argument, but if it helps you to resolve this particular objection, I'm ok with it.

 

3 hours ago, Boltzmannbrain said:

It would just be an imaginary onion that is infinitely dense with an infinite number of layers.  You peel one layer back, and you would expect a next layer, right?

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.

 

3 hours ago, Boltzmannbrain said:

Yes, I have always understood that.  My counter arguments that pop up in my head leave me unconvinced and also curious as to what it is that I still do not understand about this topic.

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. 

If you could focus on the proof we could discuss that. I can't discuss onions or bodies of water or vague pre-mathematical notions of the real numbers. It's more helpful to focus on the actual math.

 

3 hours ago, Boltzmannbrain said:

And I appreciate this a lot.  I have learnt a lot since starting this thread.

This is just a long shot, but what if we discover something?  That would be great for all of us since we have all worked on this together.

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. 

3 hours ago, Boltzmannbrain said:

Are you sure they are proofs?  If so, what kind of proofs are they? Mathematical proof - Wikipedia  I am quite familiar with most of them, but not all.  

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

There is no largest number in [0,1). 

Proof: Suppose you claim that x∈[0,1) is the largest number in that set. Take half the distance between x and 1 , namely 1−x2 and add it to x , giving:

x+1−x2

You can see that we have the strict inequality x<x+1−x2<1  so that x is not the largest number in [0,1) after all.

Since x is entirely arbitrary, we have just shown that there is no largest number in [0,1).

You have already seen this proof, and more or less said you agree with it. But now you are saying "Are you sure there are proofs?" You already saw the proof. Yes, I'm sure there's a proof, I've now stated it twice. And you've agreed to it. So I have no idea what you mean by asking if I'm sure there's a proof.

Ah ... ps ... you said, am I sure THEY are proofs. Are you asking if the proof I gave is actually a proof? Yes, I'm sure. If there is any part of it you are unsure of, I wish you would ask about it or say which part you find unconvincing, so that we can focus on that.

 

Edited March 12, 2023 by wtf

[studiot]

8 hours ago, 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?
 

 

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.

 

Ah. Well, the length of a single point is zero. The length of [0,1] is 1, and so is the length of [0,1). The addition or deletion of a single point makes no difference when we're calculating the length of an interval. 

 

 

I'll keep my clothes on if it's all the same to you, thanks.

 

The length of a point is zero, so adding or deleting a point can not make any difference in the length of a line segment.

I think what you are saying is that the intervals [0,5] and [5,6] both contain the number 5, and that is correct. I don't see how that helps you to find a largest number in [0,1). So the two intervals would "bang into each other" at the point 5. 

You are correct that 5 is an element of both intervals. But as a point, the number 5 has length 0. The two facts are both true. 5 is a point on the real number line and it has length 0. 

So yes, two points of zero length can still bang into each other, if you want to put it that way. Remember, Newton showed that you can reduce gravitational calculations to "point masses." So if it helps, you can think of them that way. They are points with zero dimensions, zero length, and zero volume, but they still pack a punch. I'm not saying that's any kind of mathematical argument, but if it helps you to resolve this particular objection, I'm ok with it.

 

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.

 

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. 

If you could focus on the proof we could discuss that. I can't discuss onions or bodies of water or vague pre-mathematical notions of the real numbers. It's more helpful to focus on the actual math.

 

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?

There is no largest number in [0,1). 

Proof: Suppose you claim that x∈[0,1) is the largest number in that set. Take half the distance between x and 1 , namely 1−x2 and add it to x , giving:

x+1−x2

You can see that we have the strict inequality x<x+1−x2<1  so that x is not the largest number in [0,1) after all.

Since x is entirely arbitrary, we have just shown that there is no largest number in [0,1).

You have already seen this proof, and more or less said you agree with it. But now you are saying "Are you sure there are proofs?" You already saw the proof. Yes, I'm sure there's a proof, I've now stated it twice. And you've agreed to it. So I have no idea what you mean by asking if I'm sure there's a proof.

Ah ... ps ... you said, am I sure THEY are proofs. Are you asking if the proof I gave is actually a proof? Yes, I'm sure. If there is any part of it you are unsure of, I wish you would ask about it or say which part you find unconvincing, so that we can focus on that.

 

Impressive and patient replies.  +1

[Genady]

3 hours ago, studiot said:

Impressive and patient replies.  +1

Yes, agree. Another +1 to @wtf.

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