Regarding Cantor's Diagonal Argument

[Juma]

So you agree that there is no contradiction?

How did you get that from what I said?

Because i interpreted the second section as an insight on how it is to think that kardinal numbers of infinite sets could be finite numbers:

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

There simply is no such natural number.
Why would there?

 

[ryan]

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.

Except, functions do not have to equal the limits they approach.

 

[ryan]

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.

Well there must be a reason for why you treat people the way you do.

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

I took first year calculus in university. We didn't cover this. Do you seriously expect me to spend more years and a lot more money just to understand this question? I love math passionately, but I have other things that I also love, and I have to balance them.

 

[Juma]

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.

Except, functions do not have to equal the limits they approach.

So what? What exactly is the point of that comment? I mean: in what way makes it your argument earlier valid?

The limit of n when n goes to infinity is infinity and that is not a natursl number.

 

[ryan]

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.

It must be a natural number because only natural numbers belong to the set of natural numbers. It seems reasonable to have the nth element equal to the value of n. But then either both are finite or both are infinite. I don't see any alternative.

 

[ryan]

Except, functions do not have to equal the limits they approach.

So what? What exactly is the point of that comment? I mean: in what way makes it your argument earlier valid?

It was meant to invalidate EricK's argument against mine.

The limit of n when n goes to infinity is infinity and that is not a natursl number.

It is not a requirement that the function equals the limit.

 

[Juma]

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.

It must be a natural number because only natural numbers belong to the set of natural numbers.

Why do you think it has to belong to the set of natural numbers?

It seems reasonable to have the nth element equal to the value of n. But then either both are finite or both are infinite. I don't see any alternative.

The n:th element is n but the limit when n goes to infinity is not a natural number.

 

[Juma]

So what? What exactly is the point of that comment? I mean: in what way makes it your argument earlier valid?

It was meant to invalidate EricK's argument against mine.

Sigh...,
Yes I get that your intention was to invalidate EricKs argument but since what you wrote doesnt seem to have anything to do with his post I had to ask for an explanation HOW it invalidates EricKs argument.

 

[ryan]

It must be a natural number because only natural numbers belong to the set of natural numbers.

Why do you think it has to belong to the set of natural numbers?

Because it is EQUAL to the natural number that it matches.

It seems reasonable to have the nth element equal to the value of n. But then either both are finite or both are infinite. I don't see any alternative.

The n:th element is n but the limit when n goes to infinity is not a natural number.

Well that's the debate. If what you say here is true, then I am wrong.

- - - Updated - - -

It was meant to invalidate EricK's argument against mine.

Sigh...,
Yes I get that your intention was to invalidate EricKs argument but since what you wrote doesnt seem to have anything to do with his post I had to ask for an explanation HOW it invalidates EricKs argument.

No sigh, you asked me how it makes my argument valid. I was only pointing out that his argument against me isn't valid.

 

[J842P]

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.

It must be a natural number because only natural numbers belong to the set of natural numbers. It seems reasonable to have the nth element equal to the value of n. But then either both are finite or both are infinite. I don't see any alternative.

But why do you think cardinalities of sets have to belong to the natural numbers? This is only the case for finite sets. It doesn't contradict anything, unless you postulate, as you have, that n must belong to the natural numbers. In which case, the cardinality of infinite sets is not defined.

- - - Updated - - -

Why do you think it has to belong to the set of natural numbers?

Because it is EQUAL to the natural number that it matches.

It seems reasonable to have the nth element equal to the value of n. But then either both are finite or both are infinite. I don't see any alternative.

The n:th element is n but the limit when n goes to infinity is not a natural number.

Well that's the debate. If what you say here is true, then I am wrong.

- - - Updated - - -

It was meant to invalidate EricK's argument against mine.

Sigh...,
Yes I get that your intention was to invalidate EricKs argument but since what you wrote doesnt seem to have anything to do with his post I had to ask for an explanation HOW it invalidates EricKs argument.

No sigh, you asked me how it makes my argument valid. I was only pointing out that his argument against me isn't valid.

No, ryan. Erik's argument doesn't say what you think it does. Why don't you lay out the assumptions and logical inferences required to get at the conclusion that n MUST be a natural number.

 

[J842P]

That's what I suspected. The issue here, is that the "number" you get when you try to apply the diagonal method isn't a natural number. Every natural number has a finite number of digits - a first digit, a second digit etc, however what you will end up with is an infinite string of digits with no beginning.

I can either have a beginning with no end or an end with no beginning. I chose to have it begin in the ones position. Just reflect the rows and they will start in the ones position.

Cantor's diagonal argument for the Reals works because a real number is an infinite string of digits with no end, and the construction produces such a number.

But the nth row and nth column can only ever be natural numbers too. So the rows will only ever have an n number of digits.

Here is the source of your error, ryan.

 

[ryan]

It must be a natural number because only natural numbers belong to the set of natural numbers. It seems reasonable to have the nth element equal to the value of n. But then either both are finite or both are infinite. I don't see any alternative.

But why do you think cardinalities of sets have to belong to the natural numbers? This is only the case for finite sets. It doesn't contradict anything, unless you postulate, as you have, that n must belong to the natural numbers. In which case, the cardinality of infinite sets is not defined.

I will wrap everything up below.

No, ryan. Erik's argument doesn't say what you think it does. Why don't you lay out the assumptions and logical inferences required to get at the conclusion that n MUST be a natural number.

Like I told Juma, n has to belong to the set of natural numbers because it is equal to the value of the natural number that it matches to.

 

[J842P]

But why do you think cardinalities of sets have to belong to the natural numbers? This is only the case for finite sets. It doesn't contradict anything, unless you postulate, as you have, that n must belong to the natural numbers. In which case, the cardinality of infinite sets is not defined.

I will wrap everything up below.

No, ryan. Erik's argument doesn't say what you think it does. Why don't you lay out the assumptions and logical inferences required to get at the conclusion that n MUST be a natural number.

Like I told Juma, n has to belong to the set of natural numbers because it is equal to the value of the natural number that it matches to.

ryan, if you are going to want to discuss mathematics then you are going to have to be able to differentiate between showing some proposition is a consequence of some other set of propositions and and merely making a claim that something is so. So far, here, you only have the latter. The only postulate that is being contradicted is that n is a natural number for all sets. Your 'contradiction' merely lets you show that this postulate is, in fact, not the case. What that has to do with your discussion with Erik or for that matter Cantor's argument I haven't the foggiest idea.

 

[Juma]

Like I told Juma, n has to belong to the set of natural numbers because it is equal to the value of the natural number that it matches to.

Lets go back to how you "define" n:

Let's just take a subset of sequential natural numbers starting at 1, {1, 2, 3, 4}. In this kind of set, there are 4 natural numbers where n = 4. We can also say that n = # of elements. But then what about the set of all naturals; n cannot equal aleph 0.

You must realize that the set of all naturals is something very different from the set {1,2,3,4} in your example.

The set of all naturals is infinite and thus there is no natural number that matches the number of elements in that set.

There is no contradiction here.

 

[ryan]

ryan, if you are going to want to discuss mathematics then you are going to have to be able to differentiate between showing some proposition is a consequence of some other set of propositions and and merely making a claim that something is so. So far, here, you only have the latter.

Because it's trivially true. It is trivially true that the nth natural number starting from 1 is the value of that natural number. Besides, all I am doing is setting a rule that seems to be allowed and seeing where it takes us.

The only postulate that is being contradicted is that n is a natural number for all sets. Your 'contradiction' merely lets you show that this postulate is, in fact, not the case. What that has to do with your discussion with Erik or for that matter Cantor's argument I haven't the foggiest idea.

I started this thread with a different angle than I am using now.

Regarding the argument with EricK, I believe he was assuming that an increasing sequence bounded/limited above by 1/3 meant that the limit 1/3 equals the actual function. But we know that is not always the case, take f(x) = 2x where x can't equal 4, while we set the limit to 8. It approaches L = 8 without ever equaling 8.

 

[J842P]

Because it's trivially true. It is trivially true that the nth natural number starting from 1 is the value of that natural number. Besides, all I am doing is setting a rule that seems to be allowed and seeing where it takes us.

But ryan, that isn't what you are stating. You are stating that n, where n is the cardinality of some set, must be a natural number, then saying that is a contradiction, and using that contradiction to show.... what exactly?

But you haven't show that the above must be the case. You've merely postulated it. And indeed, now it just seems to me that all you are saying is that the nth natural number starting from 1 is precisely n. But that has nothing to do with what you or other people have been discussing previously. I reposted the exchange with you and EricK, and your mistake is trying to treat an infinite string of digits like a natural number, which it is NOT. All natural numbers can be represented as finite strings of digits. The only conclusion I can reach is that you are fundamentally misunderstanding at least a few core ideas.

Also, not to belabor the point, but what do you think that contradiction would show anyway? Why do you think you hvae shown anything other than the fact that the cardinality of infinite sets cannot be represented by natural numbers? And if that indeed was your purpose, to what end?

 

[Juma]

Because it's trivially true. It is trivially true that the nth natural number starting from 1 is the value of that natural number. Besides, all I am doing is setting a rule that seems to be allowed and seeing where it takes us.

The only postulate that is being contradicted is that n is a natural number for all sets. Your 'contradiction' merely lets you show that this postulate is, in fact, not the case. What that has to do with your discussion with Erik or for that matter Cantor's argument I haven't the foggiest idea.

I started this thread with a different angle than I am using now.

Regarding the argument with EricK, I believe he was assuming that an increasing sequence bounded/limited above by 1/3 meant that the limit 1/3 equals the actual function. But we know that is not always the case, take f(x) = 2x where x can't equal 4, while we set the limit to 8. It approaches L = 8 without ever equaling 8.

Erick wasnt talking about a function, he was talking about a sequence and the limit of that sequence.

 

[ryan]

Lets go back to how you "define" n:

Let's just take a subset of sequential natural numbers starting at 1, {1, 2, 3, 4}. In this kind of set, there are 4 natural numbers where n = 4. We can also say that n = # of elements. But then what about the set of all naturals; n cannot equal aleph 0.

You must realize that the set of all naturals is something very different from the set {1,2,3,4} in your example.

The set of all naturals is infinite and thus there is no natural number that matches the number of elements in that set.

There is no contradiction here.

You are still just trying to force your argument to be true.

 

[J842P]

Why have you refused to lay out your argument as you see it? Something like beero or Angra Mainyu has done, that is, start with your assumptions, make your inferences explicit, and end with your conclusion?

 

[Juma]

Lets go back to how you "define" n:


You must realize that the set of all naturals is something very different from the set {1,2,3,4} in your example.

The set of all naturals is infinite and thus there is no natural number that matches the number of elements in that set.

There is no contradiction here.

You are still just trying to force your argument to be true.

Then show where I am wrong!