Confusion with finite sets vs an infinite set of natural numbers

[pzkpfw]

36 minutes ago, Boltzmannbrain said:

 

Good idea, I will remove the sets from it, especially since it doesn't seem to be getting me anywhere.

Imagine a meter of paint is painted for every natural number that is in the set of natural numbers N.  Every n is finite, so how can the paint be infinitely far away?

It isn't a proof, but do you at least see how there is something that deserves attention?

...

 

Each metre of paint is a nice finite metre.

But the line is an infinitely long line (made up of finite metres).

I really don't see any issue deserving attention.

 

[wtf]

48 minutes ago, Boltzmannbrain said:

magine a meter of paint is painted for every natural number that is in the set of natural numbers N.  Every n is finite, so how can the paint be infinitely far away?

There is no paint infinitely far away. Pick any point on the line of paint. It's a finite distance from where you started. 

It's true that the line of paint is infinitely long. But there is no point on the line that is infinitely far away from anything else on the line. 

Let the line of paint be modeled by the usual real number line. I challenge you to identify any two points on the line that are an "infinite distance" from each other.

On the contrary, if any two points on the line are labeled x and y, then their distance is the finite real number |x - y|.

Can you see that? There is never an "infinite distance" between any two points on the real number line.

Edited May 4, 2023 by wtf

[Boltzmannbrain]

2 hours ago, pzkpfw said:

 

Each metre of paint is a nice finite metre.

But the line is an infinitely long line (made up of finite metres).

I really don't see any issue deserving attention.

 

If we numbered each meter from 1 and continued to increase each consecutive meter by 1, wouldn't we have a natural number infinitely long?  It seems like we would since we would have essentially added 1 an infinite number of times for some meter. 

2 hours ago, wtf said:

There is no paint infinitely far away. Pick any point on the line of paint. It's a finite distance from where you started. 

It's true that the line of paint is infinitely long. But there is no point on the line that is infinitely far away from anything else on the line. 

Let the line of paint be modeled by the usual real number line. I challenge you to identify any two points on the line that are an "infinite distance" from each other.

On the contrary, if any two points on the line are labeled x and y, then their distance is the finite real number |x - y|.

Can you see that? There is never an "infinite distance" between any two points on the real number line.

You said that there is no paint infinitely far away, but then you say that the line of paint is infinitely long.  Did you make a mistake, or is there somehow a difference between infinitely far and infinitely long.

[pzkpfw]

6 minutes ago, Boltzmannbrain said:

If we numbered each meter from 1 and continued to increase each consecutive meter by 1, wouldn't we have a natural number infinitely long?  It seems like we would since we would have essentially added 1 an infinite number of times for some meter. 

..

Sure, why not? Why is that an issue?

Do you think we should stop at 3 billion metres?

[Boltzmannbrain]

34 minutes ago, pzkpfw said:

Sure, why not? Why is that an issue?

Do you think we should stop at 3 billion metres?

The meter numbering is analogous to the set of natural numbers.  There can't be an infinitely large natural number in the set of natural numbers.

[pzkpfw]

33 minutes ago, Boltzmannbrain said:

The meter numbering is analogous to the set of natural numbers.  There can't be an infinitely large natural number in the set of natural numbers.

So what is the "largest" natural number in the set of natural numbers?

[Boltzmannbrain]

15 minutes ago, pzkpfw said:

So what is the "largest" natural number in the set of natural numbers?

There is no largest natural number, and none of them can be infinite.

[pzkpfw]

36 minutes ago, Boltzmannbrain said:

There is no largest natural number, and none of them can be infinite.

Yes, there's no largest natural number, because there's infinite of them, yet no individual number is infinite.

Infinity is weird like that. I'm out for now.

[Boltzmannbrain]

2 minutes ago, pzkpfw said:

Yes, there's no largest natural number, because there's infinite of them, yet no individual number is infinite.

Infinity is weird like that. I'm out for now.

But what about a solution to my analogy?

[wtf]

3 hours ago, Boltzmannbrain said:

You said that there is no paint infinitely far away, but then you say that the line of paint is infinitely long.  Did you make a mistake, or is there somehow a difference between infinitely far and infinitely long.

No mistake, this is exactly the point you seem to be having trouble with.

What is infinitely far? I challenge you to give me any two numbers that are "infinitely far" apart.

On the contrary, if you pick any two real numbers, the distance between them is always finite.

Can you see that? Try some examples. 10 and 47. Distance is 37. Try the distance between 0 and googolplex. Distance is googolplex, a finite number.

Can you see that even though the real number line is infinitely long, the distance between any two points on it is finite? 

It's imperative that you see this. Try some examples for yourself.

You cannot find any two numbers that are "infinitely far" from one another.

Can you see this?

Edited May 4, 2023 by wtf

[pzkpfw]

1 hour ago, Boltzmannbrain said:

But what about a solution to my analogy?

Solution for what? I fail to see anything that needs a solution!

(That's why I'm out for now. This is clearly going nowhere.)

[Boltzmannbrain]

10 hours ago, wtf said:

No mistake, this is exactly the point you seem to be having trouble with.

What is infinitely far? I challenge you to give me any two numbers that are "infinitely far" apart.

 

Infinitely far would be a distance, of say 2 objects, that are infinitely far from each other.  No natural number n can measure it.  What is wrong with that?

2 numbers within a finite distance on a number line is always going to be a finite distance. 

  

Quote

 

On the contrary, if you pick any two real numbers, the distance between them is always finite.

Can you see that? Try some examples. 10 and 47. Distance is 37. Try the distance between 0 and googolplex. Distance is googolplex, a finite number.

Can you see that even though the real number line is infinitely long, the distance between any two points on it is finite? 

It's imperative that you see this. Try some examples for yourself.

You cannot find any two numbers that are "infinitely far" from one another.

Can you see this?

 

 

I agree, but I don't see how this helps solve my issue.

 

9 hours ago, pzkpfw said:

Solution for what? I fail to see anything that needs a solution!

(That's why I'm out for now. This is clearly going nowhere.)

 

Solution to my post.  You haven't addressed it.  

 

Quote

The meter numbering is analogous to the set of natural numbers.  There can't be an infinitely large natural number in the set of natural numbers.

 

I will reword it.  There would have to exist an infinitely long natural number in order to label every meter of paint. 

 

 

Edited May 4, 2023 by Boltzmannbrain

[wtf]

1 hour ago, Boltzmannbrain said:

I agree, but I don't see how this helps solve my issue.

If you agree with what I wrote but don't seen how this solves your issue, then I do not understand your issue. 

 

1 hour ago, Boltzmannbrain said:

There would have to exist an infinitely long natural number in order to label every meter of paint. 

How do you figure that? Name a meter of paint that can't be labeled with a normal finite-length natural number.

Likewise, name a point on the real number line that can't be labelled with a finite real number.

What kind of line are you imagining that isn't the real number line?

[Boltzmannbrain]

52 minutes ago, wtf said:

How do you figure that? Name a meter of paint that can't be labeled with a normal finite-length natural number.

 

 

You can't get to an infinitely long distance with any finite number/distance.  There is always more than any finite number/distance.

 

Quote

Likewise, name a point on the real number line that can't be labelled with a finite real number.

 

Imagine there is a point a distance of 1 away from an origin 0.  Then imagine we pushed the point away by 1 an infinite (of size aleph null) amount of times.  What n would measure from 0 to that point?

 

[studiot]

3 hours ago, Boltzmannbrain said:

I agree, but I don't see how this helps solve my issue.

 

3 hours ago, Boltzmannbrain said:

There would have to exist an infinitely long natural number in order to label every meter of paint. 

 

Yes this is one place you are most definitely going astray.

Please note this does not mean I consider you to be in any way an idiot.

 

An infinite line does not have a length, any more than N has a greatest or last (END) element.

 

In order for a line to have a length it must have two ends. A beginning and a termination.

The natural number line has a beginning but no end (termination)

The integer line, the rational number line and the real number line have neither beginning nor end (termination).

 

 

 

[wtf]

19 minutes ago, Boltzmannbrain said:

Imagine there is a point a distance of 1 away from an origin 0.  Then imagine we pushed the point away by 1 an infinite (of size aleph null) amount of times.  What n would measure from 0 to that point?

We don't usually think of transfinite cardinals as measuring distance. They measure quantity.

But even though I understand this point, I don't see the heart of your objection. Sure, in the extended real number system there is an infinite distance between 0 and the point at infinity. But this doesn't really mean anything in terms of what you 'are saying.

And especially, going back to your original question, about the R(n)'s. Each set {1, 2, 3, ..., n} is a finite set, and there are infinitely many such sets. There is just no mystery there.

[Boltzmannbrain]

1 hour ago, studiot said:

 

 

Yes this is one place you are most definitely going astray.

Please note this does not mean I consider you to be in any way an idiot.

 

An infinite line does not have a length, any more than N has a greatest or last (END) element.

 

In order for a line to have a length it must have two ends. A beginning and a termination.

The natural number line has a beginning but no end (termination)

The integer line, the rational number line and the real number line have neither beginning nor end (termination).

 

I suppose what I am really saying in this thread is that enumerating infinity (with the cardinality of aleph null) to exhaustion with the finite property of all natural numbers is not logical.  Any n you choose will not be infinite like infinity is.  That means that there is no finite n that can reach infinity. 

 

1 hour ago, wtf said:

We don't usually think of transfinite cardinals as measuring distance. They measure quantity.

But even though I understand this point, I don't see the heart of your objection. Sure, in the extended real number system there is an infinite distance between 0 and the point at infinity. But this doesn't really mean anything in terms of what you 'are saying.

 

Okay, distance is not a necessary part of my objection.  We can discuss quantity instead.  The question that I asked you in the post you are replying to here can be altered to say: what n would quantify the intervals from 0 to the point that was pushed away?
 

Quote

 

And especially, going back to your original question, about the R(n)'s. Each set {1, 2, 3, ..., n} is a finite set, and there are infinitely many such sets. There is just no mystery there.

 

 

 

 

I have narrowed my confusion a lot since the OP to what I said above in this post. 

Edited May 4, 2023 by Boltzmannbrain
I took out an incomplete thought/sentence.

[wtf]

37 minutes ago, Boltzmannbrain said:

Any n you choose will not be infinite like infinity is.  That means that there is no finite n that can reach infinity. 

Yes ok, but who said there was one? The process 1, 2, 3, 4, ... is endless. Infinite if you like. It never ends. So we say that the quantity of natural numbers is infinite. Why does this trouble you? There is no "last" number right before infinity, there's just the endless sequence 1, 2, 3, ... Or in set theory, we can collect all of them into a set: {1, 2, 3, 4, ...}. What of it? Why is this troubling you?

 

37 minutes ago, Boltzmannbrain said:

The question that I asked you in the post you are replying to here can be altered to say: what n would quantify the intervals from 0 to the point that was pushed away?

If it was pushed away to point 47, the distance would be 47. No matter where you push it away to, the distance is finite. 

If you conceptualize a "point at infinity," as is done in the extended real number system, then the distance is infinity. But again, what about this is troubling you?

[Boltzmannbrain]

1 hour ago, wtf said:

Yes ok, but who said there was one? The process 1, 2, 3, 4, ... is endless. Infinite if you like. It never ends. So we say that the quantity of natural numbers is infinite. Why does this trouble you?

 

Each natural number has to be finite, right?

 

Quote

 

There is no "last" number right before infinity, there's just the endless sequence 1, 2, 3, ... Or in set theory, we can collect all of them into a set: {1, 2, 3, 4, ...}. What of it? Why is this troubling you?

 

If it was pushed away to point 47, the distance would be 47. No matter where you push it away to, the distance is finite. 

If you conceptualize a "point at infinity," as is done in the extended real number system, then the distance is infinity. But again, what about this is troubling you?

 

 

For the same reason I mentioned above in this post.  

 

[Boltzmannbrain]

4 hours ago, wtf said:

If it was pushed away to point 47, the distance would be 47. No matter where you push it away to, the distance is finite. 

If you conceptualize a "point at infinity," as is done in the extended real number system, then the distance is infinity. But again, what about this is troubling you?

 

There must be an n that can reach that point.  That n would have to be infinite, right?

[studiot]

14 hours ago, Boltzmannbrain said:

I suppose what I am really saying in this thread is that enumerating infinity (with the cardinality of aleph null) to exhaustion with the finite property of all natural numbers is not logical.  Any n you choose will not be infinite like infinity is.  That means that there is no finite n that can reach infinity. 

Most definitely, +1

 

So why do you keep trying to do it ?

(I didn't say -1 here because I don't give negative points.)

 

Let us look more closely at the famous painted line of yours.

How wide is it ?

9 hours ago, Boltzmannbrain said:

There must be an n that can reach that point.  That n would have to be infinite, right?

 

Why ever should you think this ?

 

There are whole infinities of points that to use that ugly phrase, "that cannot be reached "

For instance every point in the plane alongside your famous painted line.

 

How long is this painted line ?

Consider three cases ?

You paint the first metre segment and chalk up n = 1

You paint the second metre segment and chalk up n = 2

and so on. A German Mathmatician would say USW  = und so weiter

 

BUT

You paint the first 1/1  metre segment and chalk up n = 1

You paint the second 1/2 metre segment and chalk up n = 2

You paint the third 1/3 metre ssegment and chalk up n = 3

USW

 

OR

You paint the first 1/1  metre segment and chalk up n = 1

You paint the second 1/4 metre segment and chalk up n = 2

You paint the third 1/9 metre segment and chalk up n = 3

 

Which strip is the longest ?

Can you say anything else about the lengths of these strips ?

You can say they all have exactly the same count of segments.

[Boltzmannbrain]

4 hours ago, studiot said:

Quote

There must be an n that can reach that point.  That n would have to be infinite, right?

Why ever should you think this ?

 

 

Here is the Boltzmannbrain Line (all rights reserved) that ends with an X

 

1,   2,   3,   4,   .   .   .   infinity (countably infinite)   X,

We are using only natural numbers to count the commas on the line in order to get to that X

There are an infinite number of commas to get to the X.

There must be an n that is infinitely large to count the X's comma.

 

Quote

 

There are whole infinities of points that to use that ugly phrase, "that cannot be reached "

For instance every point in the plane alongside your famous painted line.

 

How long is this painted line ?

Consider three cases ?

You paint the first metre segment and chalk up n = 1

You paint the second metre segment and chalk up n = 2

and so on. A German Mathmatician would say USW  = und so weiter

 

BUT

You paint the first 1/1  metre segment and chalk up n = 1

You paint the second 1/2 metre segment and chalk up n = 2

You paint the third 1/3 metre ssegment and chalk up n = 3

USW

 

OR

You paint the first 1/1  metre segment and chalk up n = 1

You paint the second 1/4 metre segment and chalk up n = 2

You paint the third 1/9 metre segment and chalk up n = 3

 

Which strip is the longest ?

Can you say anything else about the lengths of these strips ?

You can say they all have exactly the same count of segments.

 

I am not sure why we have to get into all of this.  The heart of my issue is above.

Edited May 5, 2023 by Boltzmannbrain

[studiot]

23 minutes ago, Boltzmannbrain said:

that ends with an X

How many times do you have to be told by several differe

nt members in several different ways.

 

There is no end to the line.

There is no last number, symbol, hobgoblin or anything else.  (what comes after X ?)

Infinity is not only not a number it is not a member of the natural numbers, N, the rational numbers, Q or the Real numbers R.

There is no point at infinity in any of these systems.

All of these statements are equivalent and mean that to 'reach infinity' you have to step outside the existing system and establish a new one.

If you do this you have to prove that all the existing rules and relationships hold good in your new system, you can't just assume them and carry on as though they apply.

 

You have definitely not done any of this spadework, yet you complain bitterly that we are hiding something from you.

Every time I ask some more searching questions about your system dodge the issue and offer this reply

31 minutes ago, Boltzmannbrain said:

I am not sure why we have to get into all of this.  The heart of my issue is above.

Your flawed line is not the 'heart of the issue'  - We need to go into all this because the examples demonstrate where the flaws lie.

When you have accepted those the process of what to do about them can commence.

 

The simplest answer is that you don't have to worry about the end, because the process does not end.

I'm sorry but if you want more you will have to do more maths. There is no other way.

For instance in my three examples, the first two will never produce a resultant length, but the last one does so a study of this difference is highly relevant.

[Boltzmannbrain]

1 hour ago, studiot said:

How many times do you have to be told by several differe

nt members in several different ways.

 

There is no end to the line.

There is no last number, symbol, hobgoblin or anything else.  (what comes after X ?)

Infinity is not only not a number it is not a member of the natural numbers, N, the rational numbers, Q or the Real numbers R.

There is no point at infinity in any of these systems.

All of these statements are equivalent and mean that to 'reach infinity' you have to step outside the existing system and establish a new one.

If you do this you have to prove that all the existing rules and relationships hold good in your new system, you can't just assume them and carry on as though they apply.

I am trying to use logic from the existing system to show why it needs an infinite n (or to understand why I am wrong). 

I want to map every n from the set of natural numbers inclusively between the points shown on the finite line below from the origin to the X.  I want to do this by us imagining that there are points on the line of every rational number between the origin and X. 

 

                                                   Origin ._______________________________________________. X  

 

There is an aleph null infinity of rational numbers inclusively between the two points of the origin and the X.  Since that I am mapping an n of N to all rational numbers, wouldn't there have to be an natural number at the point of the X?  

 

Another contradictory result is that if we zoom in to these enumerated rational points on the line, it seems to give us a first rational segment which I don't think is allowed.

 

Origen .        1.        2.        3.        4.        .        .        .   (aleph null dots) . X

 

This all must be wrong, but I don't know why.

 

Quote

 

You have definitely not done any of this spadework, yet you complain bitterly that we are hiding something from you.

Every time I ask some more searching questions about your system dodge the issue and offer this reply

Your flawed line is not the 'heart of the issue'  - We need to go into all this because the examples demonstrate where the flaws lie.

When you have accepted those the process of what to do about them can commence.

 

The simplest answer is that you don't have to worry about the end, because the process does not end.

I'm sorry but if you want more you will have to do more maths. There is no other way.

For instance in my three examples, the first two will never produce a resultant length, but the last one does so a study of this difference is highly relevant.

 

 

Okay, I will try to address what you said in the other post.  But I was confused by some of it, so I just wanted to focus on the part of the argument that we were both understanding.

 

 

Quote

 

How long is this painted line ?

Consider three cases ?

You paint the first metre segment and chalk up n = 1

You paint the second metre segment and chalk up n = 2

and so on. A German Mathmatician would say USW  = und so weiter

 

BUT

You paint the first 1/1  metre segment and chalk up n = 1

You paint the second 1/2 metre segment and chalk up n = 2

You paint the third 1/3 metre ssegment and chalk up n = 3

USW

 

 

For the second part, I think you are saying that I am chalking the numbers for the series 1/n meters.  Is that correct?  And why did you put "BUT"?  I definitely feel like I am  missing something here.

 

Quote

 

OR

You paint the first 1/1  metre segment and chalk up n = 1

You paint the second 1/4 metre segment and chalk up n = 2

You paint the third 1/9 metre segment and chalk up n = 3

 

 

Okay, but why are we doing this.  I am not getting your point.

 

Quote

 

Which strip is the longest ?

Can you say anything else about the lengths of these strips ?

You can say they all have exactly the same count of segments.

 

 

I understand what you are saying here, but I still don't understand how this helps resolve my issue.

 

 

 

 

 

 

 

Edited May 5, 2023 by Boltzmannbrain
Make it more clear

[wtf]

3 hours ago, Boltzmannbrain said:

Here is the Boltzmannbrain Line (all rights reserved) that ends with an X

 

1,   2,   3,   4,   .   .   .   infinity (countably infinite)   X,

We are using only natural numbers to count the commas on the line in order to get to that X

There are an infinite number of commas to get to the X.

There must be an n that is infinitely large to count the X's comma.

You've discovered the ordinal numbers! The first transfinite ordinal is called [math]\omega[/math], the lower-case Greek letter omega. It's a number that "comes after" all the finite natural numbers.

In set theory it's exactly the same set as [math]\aleph_0[/math] but considered as an ordinal (representing order) rather than a cardinal (representing quantity).

So the ordinal number line begins:

0, 1, 2, 3, 4, ... [math]\omega[/math], ...

Now the point is, there is no "last" natural number [math]n[/math] that "reaches" or "is right before" [math]\omega[/math]. It doesn't work that way. If you are at [math]\omega[/math] and you take a step backwards, you will land on some finite natural number. But there are still infinitely many other natural numbers to the right of the one you landed on.

You can jump back from [math]\omega[/math] to some finite natural number (which still has infinitely many natural numbers after it), but you can't jump forward a single step to get back to [math]\omega[/math].

That's just how it works. 

There's even a technical condition that lets us recognize why [math]\omega[/math] is special.

A successor ordinal is an ordinal that has an immediate predecessor. All the finite natural numbers except 0 are successor ordinals.

A limit ordinal is an ordinal that has no immediate predecessor. [math]\omega[/math] is a limit ordinal. That is, there is no other ordinal whose successor is [math]\omega[/math].

Note also that by this definition, 0 is also a limit ordinal. It's the only finite limit ordinal.

 

Edited May 5, 2023 by wtf