Regarding Cantor's Diagonal Argument

[EricK]

In the first cases, n is the natural number which is the highest number in the set. In the last case that definition no longer holds (as there isn't a highest number) but you haven't defined what n is.

I found an equivalence to what n is. See what I put in bold: n is also the number of elements. The number of elements in the set of the naturals is aleph 0. n Cannot equal aleph 0.

Sure. For each of the finite sets, the size of the set is n and is equal to the highest number in it. For the infinite set, the size is Aleph 0 and is not equal to any member of the set. But so what?

 

[ryan]

I found an equivalence to what n is. See what I put in bold: n is also the number of elements. The number of elements in the set of the naturals is aleph 0. n Cannot equal aleph 0.

Sure. For each of the finite sets, the size of the set is n and is equal to the highest number in it. For the infinite set, the size is Aleph 0 and is not equal to any member of the set. But so what?

The problem is that n has to equal an infinite number for there to be an infinite number of sequential natural numbers.

There clearly is a problem here.

 

[EricK]

Sure. For each of the finite sets, the size of the set is n and is equal to the highest number in it. For the infinite set, the size is Aleph 0 and is not equal to any member of the set. But so what?

The problem is that n has to equal an infinite number for there to be an infinite number of sequential natural numbers.

There clearly is a problem here.

Since n is just the size of the set, what is the problem? There are an infinite number of natural numbers.

 

[ryan]

The problem is that n has to equal an infinite number for there to be an infinite number of sequential natural numbers.

There clearly is a problem here.

Since n is just the size of the set, what is the problem? There are an infinite number of natural numbers.

And the size is aleph 0. That means that n is aleph 0, but n is only a natural number.

 

[J842P]

Since n is just the size of the set, what is the problem? There are an infinite number of natural numbers.

And the size is aleph 0. That means that n is aleph 0, but n is only a natural number.

In this case, n is not a natural number. You seem to think that since the cardinality of finite sets can be natural numbers, that the cardinality of all sets must be natural numbers.

- - - Updated - - -

Infinity is not a number. 'Infinite' is the word used to describe the size of the set of all natural numbers (and any larger sets). It is easy to see that this can not be any natural number. eg we know it is not the number 1,000 because the set {1,2,3,...,999,1000,1001} is larger than 1,000 and the size of the set of all numbers will be larger still. And we knoe it is not 1,000,000 because the set {1,2,..., 1000001} is larger. And by the same argument we can show that the size of the set of natural numbers, whatever it is, is not a natural number.

Again, this might be a situation where you are relying on your intuition about numbers, rather than just looking at the logical arguments and see where they lead.

I definitely see your logical argument, but what about my logical argument?

The set of all natural numbers can only have natural numbers in it, and each natural number represents at least that many elements. An infinite number of numbers would have to mean that a natural number would represent at least that many elements.

Clearly there is no possible natural number that can represent an infinite number of elements.

Here. Here is your error.

 

[J842P]

You can say that every natural number, n, is matched to the finite set {1,2,3...,n}. But there is no reason to assume that therefore there must be a natural number which matches to the infinite set {1,2,3...}. In fact there can't be, because we have matched each natural number to a finite set.

Unless I am misunderstanding something here, it seems like you are agreeing with me.

Yes. It just doesn't imply what you think it implies.

 

[ryan]

And the size is aleph 0. That means that n is aleph 0, but n is only a natural number.

In this case, n is not a natural number.

But each and every n must be a natural number in the set of natural numbers.

You seem to think that since the cardinality of finite sets can be natural numbers, that the cardinality of all sets must be natural numbers.

No, I was pointing out the contradiction.

 

[beero1000]

Is it really too much to ask for you to get a basic understanding of the topic before you start pointing out 'contradictions'?

Seriously, which do you think is more likely - that you've really found a catastrophic internal self-contradiction in a century-old branch of mathematics, or that you don't know enough to even grasp the explanations of why your idea doesn't work?

Just saying, you might need to learn to crawl before you try to run.

 

[ryan]

Is it really too much to ask for you to get a basic understanding of the topic before you start pointing out 'contradictions'?

Not only have you turned into a waste of time and a waste of space on this forum, but worst of all, you try to belittle people. And you want to teach?! Yikes!!!

 

[beero1000]

Is it really too much to ask for you to get a basic understanding of the topic before you start pointing out 'contradictions'?

Not only have you turned into a waste of time and a waste of space on this forum, but worst of all, you try to belittle people. And you want to teach?! Yikes!!!

Actually, I do teach, and I teach very well. More importantly, most of my students want to learn and know how to learn. They study concepts in the order I present them because I know the material, and I know which topics build on others and are important to understand. The ones that don't do that will often fail. If they don't do well because they lack a foundation of understanding, I tell them that. I'll tell them if they don't master the earlier material, they will probably fail. It might seem blunt, but it's really important to address gaps in understanding, especially if later topics depend on those ideas. I'll often put in extra work with a student who has gaps in their knowledge, and filling in those gaps usually helps them significantly. The ones that put in the effort rarely fail.

At this point, a minimum of four people have tried to help you with this, multiple times each. I'm being blunt here too because it has become abundantly clear by now that you have significant gaps in your knowledge. Basic set operations, bijections, the natural numbers, infinity, etc, are all issues you have difficulty with. Pointing that out is not belittling you. Missing some prerequisite knowledge is not a flaw and I will never tell anyone that they can't learn something. Everyone I have ever met is smart enough to learn this material and so are you. However, in preparation, you might need to learn more basic stuff first. If you don't fully understand the basics of the subject you are studying, you will probably fail. It's just as true for you as it is for my students.

Maybe saying that means I'm being a jerk. If I am, it's because being a jerk can sometimes be a pedagogically sound way to get someone to reevaluate the way they are approaching their studies. So, let me ask again: Take a step back and consider which situation is more likely - that you have destroyed a hundred-year-old piece of seminal mathematics or that you are lacking some understanding on the topic. Consider the fact that common responses to your posts in this thread are some variation of "I don't understand what you mean" or "Which part don't you understand?" and keep in mind that these are people who understand the topic and are trying to explain it to you. Think about how many explanations of the same idea have been written in the past 90 posts that had little to no effect.

Understanding is hard work, and sometimes you need take the time to work on things that you would rather skip. You need to work on them because they are necessary to accomplish the goal of learning the object of your main focus. Maybe you've skipped ahead too far. Maybe you need to learn to crawl before you can run. Maybe I'm full of shit. You're the only one who can decide. In any case, is this thread accomplishing what you wanted it to?

 

[ryan]

Not only have you turned into a waste of time and a waste of space on this forum, but worst of all, you try to belittle people. And you want to teach?! Yikes!!!

Actually, I do teach, and I teach very well. More importantly, most of my students want to learn and know how to learn. They study concepts in the order I present them because I know the material, and I know which topics build on others and are important to understand. The ones that don't do that will often fail. If they don't do well because they lack a foundation of understanding, I tell them that. I'll tell them if they don't master the earlier material, they will probably fail. It might seem blunt, but it's really important to address gaps in understanding, especially if later topics depend on those ideas. I'll often put in extra work with a student who has gaps in their knowledge, and filling in those gaps usually helps them significantly. The ones that put in the effort rarely fail.

At this point, a minimum of four people have tried to help you with this, multiple times each. I'm being blunt here too because it has become abundantly clear by now that you have significant gaps in your knowledge. Basic set operations, bijections, the natural numbers, infinity, etc, are all issues you have difficulty with. Pointing that out is not belittling you. Missing some prerequisite knowledge is not a flaw and I will never tell anyone that they can't learn something. Everyone I have ever met is smart enough to learn this material and so are you. However, in preparation, you might need to learn more basic stuff first. If you don't fully understand the basics of the subject you are studying, you will probably fail. It's just as true for you as it is for my students.

Maybe saying that means I'm being a jerk. If I am, it's because being a jerk can sometimes be a pedagogically sound way to get someone to reevaluate the way they are approaching their studies. So, let me ask again: Take a step back and consider which situation is more likely - that you have destroyed a hundred-year-old piece of seminal mathematics or that you are lacking some understanding on the topic. Consider the fact that common responses to your posts in this thread are some variation of "I don't understand what you mean" or "Which part don't you understand?" and keep in mind that these are people who understand the topic and are trying to explain it to you. Think about how many explanations of the same idea have been written in the past 90 posts that had little to no effect.

Understanding is hard work, and sometimes you need take the time to work on things that you would rather skip. You need to work on them because they are necessary to accomplish the goal of learning the object of your main focus. Maybe you've skipped ahead too far. Maybe you need to learn to crawl before you can run. Maybe I'm full of shit. You're the only one who can decide. In any case, is this thread accomplishing what you wanted it to?

So then is the place where you vent? Do you tell us what you really want to tell your students? Or, was this how you were treated, and you need to get even with the world by passing it forward?

Anyways, in every discussion with you, it is painful and usually ends with something insulting. You are way out of control with how you treat people.

 

[Juma]

Sorry ryan,but J842P gave a very clear explanation why you are wrong. Please consider that post some more.

You're error is not to recognize that you make a leap from "size of finite sets" to "size of infinite sets".

 

[ryan]

Sorry ryan,but J842P gave a very clear explanation why you are wrong. Please consider that post some more.

You're error is not to recognize that you make a leap from "size of finite sets" to "size of infinite sets".

I see the other side of the argument too. But I also do not see the logic of labeling a natural number for each of an infinite number of elements. At some point a natural number n will have to equal aleph 0 if we want to match all of the elements. But of course we can't because n can only be a natural number.

 

[Juma]

Sorry ryan,but J842P gave a very clear explanation why you are wrong. Please consider that post some more.

You're error is not to recognize that you make a leap from "size of finite sets" to "size of infinite sets".

I see the other side of the argument too. But I also do not see the logic of labeling a natural number for each of an infinite number of elements. At some point a natural number n will have to equal aleph 0 if we want to match all of the elements.

Which point? Such a point would be the upper limit of the set but there is no such upper limit.

 

[ryan]

I see the other side of the argument too. But I also do not see the logic of labeling a natural number for each of an infinite number of elements. At some point a natural number n will have to equal aleph 0 if we want to match all of the elements.

Which point? Such a point would be the upper limit of the set but there is no such upper limit.

Yes, I agree; that's what I am saying. There can't be a point where n is infinite because of what you say and also because n is a natural number.

A natural number n would have to equal aleph 0 if it is going to completely match an infinite set of numbers.

 

[Juma]

Which point? Such a point would be the upper limit of the set but there is no such upper limit.

Yes, I agree; that's what I am saying. There can't be a point where n is infinite because of what you say and also because n is a natural number.

A natural number n would have to equal aleph 0 if it is going to completely match an infinite set of numbers.

So you agree that there is no contradiction?

 

[EricK]

I think your error comes in thinking that what is true of each member of a sequence must also be true of the limit of that sequence. But this is false.

Consider the sequence 0.3, 0.33, 0.333, 0.3333 etc. Each number in this sequence is strictly less than 1/3. But the limit of this sequence is exactly equal to 1/3. So what is true of every member might be false for the limit.

So with the sets {1}, {1,2}, {1,2,3} etc. Each of them has a size, n, which is a natural number. But the limit of that sequence is the set of all natural numbers and that doesn't have a size which is a natural number.

 

[beero1000]

Actually, I do teach, and I teach very well. More importantly, most of my students want to learn and know how to learn. They study concepts in the order I present them because I know the material, and I know which topics build on others and are important to understand. The ones that don't do that will often fail. If they don't do well because they lack a foundation of understanding, I tell them that. I'll tell them if they don't master the earlier material, they will probably fail. It might seem blunt, but it's really important to address gaps in understanding, especially if later topics depend on those ideas. I'll often put in extra work with a student who has gaps in their knowledge, and filling in those gaps usually helps them significantly. The ones that put in the effort rarely fail.

At this point, a minimum of four people have tried to help you with this, multiple times each. I'm being blunt here too because it has become abundantly clear by now that you have significant gaps in your knowledge. Basic set operations, bijections, the natural numbers, infinity, etc, are all issues you have difficulty with. Pointing that out is not belittling you. Missing some prerequisite knowledge is not a flaw and I will never tell anyone that they can't learn something. Everyone I have ever met is smart enough to learn this material and so are you. However, in preparation, you might need to learn more basic stuff first. If you don't fully understand the basics of the subject you are studying, you will probably fail. It's just as true for you as it is for my students.

Maybe saying that means I'm being a jerk. If I am, it's because being a jerk can sometimes be a pedagogically sound way to get someone to reevaluate the way they are approaching their studies. So, let me ask again: Take a step back and consider which situation is more likely - that you have destroyed a hundred-year-old piece of seminal mathematics or that you are lacking some understanding on the topic. Consider the fact that common responses to your posts in this thread are some variation of "I don't understand what you mean" or "Which part don't you understand?" and keep in mind that these are people who understand the topic and are trying to explain it to you. Think about how many explanations of the same idea have been written in the past 90 posts that had little to no effect.

Understanding is hard work, and sometimes you need take the time to work on things that you would rather skip. You need to work on them because they are necessary to accomplish the goal of learning the object of your main focus. Maybe you've skipped ahead too far. Maybe you need to learn to crawl before you can run. Maybe I'm full of shit. You're the only one who can decide. In any case, is this thread accomplishing what you wanted it to?

So then is the place where you vent? Do you tell us what you really want to tell your students? Or, was this how you were treated, and you need to get even with the world by passing it forward?

Anyways, in every discussion with you, it is painful and usually ends with something insulting. You are way out of control with how you treat people.

Yes, yes, I am a cruel and petty man who is desperate to fill the hole in my heart that was created when someone was mean to me.

Now that we've established that, have you put any thought into the actual *substance* of my post?

 

[ryan]

Yes, I agree; that's what I am saying. There can't be a point where n is infinite because of what you say and also because n is a natural number.

A natural number n would have to equal aleph 0 if it is going to completely match an infinite set of numbers.

So you agree that there is no contradiction?

How did you get that from what I said? I am saying here that n is a natural number but then it is also an infinite number.

 

[J842P]

So you agree that there is no contradiction?

How did you get that from what I said? I am saying here that n is a natural number but then it is also an infinite number.

Again, your mistake is thinking n must be a natural number. Why do you think this is true? Just because it is called 'n' doesn't mean it has to come from the natural numbers.