2 Questions about Cantor's Diagonal Argument

[Mates]

First, there is no need to “create“ the decimal representation of 1/3 to ADD to the list of the decimal representations of every rational number. It is ALREADY there one time by definition. With me on that?

No I am not. Because, why wouldn't we just say that with the original argument of the reals?

Second, you are making diagonalization unnecessatily complex as well as missing the point.

I know. That is why I am here. The fact that I have come up with the example that we are discussing, already tells me that I have a misunderstanding of the argument or that my understanding of it is incomplete.

If, for example, 1/3‘s expansion is the 1715th expansion in the list, ...

I am already lost. I do not understand what you are saying. I posted a diagram of what I am trying to convey.

then our creation of our new number insets any digit except 3 or 9 as the 1715th digit to the right of the decimal point in the new number being constructed. Therefore, that number will not equal 1/3: it will not be all 3’s by construction. Of course, it could be identical to the expansion of 1/3 in every other digit up to that one, but it will not equal 1/3 because it differs by at least one digit.

Moreover, that number being constructed will not equal the first number in the list because the number being constructed will differ from the first number in the list by at least the first digit to the right of the decimal point. It will not equal the the second number in the list because it will differ from the second number by at least the second digit to the right of the decimal point. In short, the new number constructed cannot be in the list. This is true whether diagonalization is applied to the list of all rational numbers or the hypothetical list of all real numbers.

You seem to have persuaded yourself that we cannot apply the diagonalization process to the list of rational numbers. This is simply not true as I hope you now realize.

So, the answer to your fundamental question that you started out with (Why can’t we apply the diagonalization process to the list of the decimal expansion of all rational numbers) is that you CAN DO THAT, and you get the SAME IMMEDIATE RESULT, namely the specification of a real number that is not in the list.

It is the CONSEQUENCE of that immediate result that differs between rational and real numbers. There is no contradiction in finding a real number that is necessarily not in the list of all rationals; there are real numbers that are not rational numbers, namely the irrationals. There is a contradiction in finding a real number that necessarily is not in the purported list of all real numbers.

I doubt I can make the argument clearer so I hope I have succeeded this time.

There must be miscommunication somewhere (probably by my part), because I do not even understand what you are saying or why you are saying it.

 

[blamocur]

Now I am confused even more: if you choose "not 3" how do you get all 3's ? If you wanted to make 0.3333... "not listable" ("not listed"?) you would have to choose 3 in each location of the diagonal, which you cannot in the general case.
Am I missing something here?

 

[NotJeffM]

I am trying to understand the significance of Cantor's diagonal argument. Here are 2 questions just to give an example of my confusion.

From what I understand so far about the diagonal argument, it finds a real number that cannot be listed in any nth row, as n (from the set of natural numbers) goes to infinity. It does this by listing real numbers and making a rule that its nth column cannot have a digit in the corresponding nth row. This produces a real number that cannot be listed using the countably infinite (aleph null) set of rows. It is said that the reason this number can be created and not listed is because the reals are a set that is even larger than the aleph null set.

[/QUOTE]
You say “a rule that [a listed number’s] nth column [actually you mean digit to the right of the decimal point] cannot have a digit.” That literally makes no sense. There is a list of numbers in decimal form, each number containing an infinite number of digits to the right of the decimal point (excluding infinite 9’s). What Cantor’s diagonalization procedure does is to construct a number that is not in the list. It is not in the list because it is specifically constructed to differ by at least digit from each number in the list.

Here are the parts that I do not understand. I do not understand why we can't just make the same argument using the rationals instead of reals.

Question 1: I know the rationals have a one-to-one correlation with the naturals and thus the same cardinality, wouldn't the diagonal argument work the same way that it does for the reals?

[/QUOTE]
If what you are asking is whether we can construct a real number that is not in the list of all rational numbers, the answer is that we can.

In your most recent thread, you ask why can we not make the number not in the list be the decimal representation of 1/3. We can’t because the list of all rational numbers certainly includes 1/3 already by definition.

For example, since we can make 1/3 be the number not "listable" (by choosing not 3 in the diagonal direction after the decimal point) doesn't this mean that we falsely proved that the set of rationals is larger than the set of the naturals?

[/QUOTE]
No we cannot make 1/3 be the number not listable in the list of rational numbers.

Question 2: Instead of listing all of the naturals to a list of reals, couldn't we just list every 2nd n or n^2 so that there are always more naturals left to correlate to any reals that can't be correlated in the original list?

I am really confused and curious how the diagonal argument handles questions like these.

[/QUOTE]
What is your procedure to create this listing?

 

[Mates]

Now I am confused even more: if you choose "not 3" how do you get all 3's ? If you wanted to make 0.3333... "not listable" ("not listed"?) you would have to choose 3 in each location of the diagonal, which you cannot in the general case.
Am I missing something here?

The way I understand it is that in the argument (from Wikipedia), in base 2 for example, the rule is essentially either "not 0" or "not 1" at every diagonal position. With my example, I am just excluding 3 of the 9 other choices.

 

[blamocur]

The way I understand it is that in the argument (from Wikipedia), in base 2 for example, the rule is essentially either "not 0" or "not 1" at every diagonal position. With my example, I am just excluding 3 of the 9 other choices.

Still confused: if you are excluding 3's how do you get 0.3333... ?

 

[Mates]

What Cantor’s diagonalization procedure does is to construct a number that is not in the list. It is not in the list because it is specifically constructed to differ by at least digit from each number in the list.

That is what I am trying to do too? I am trying to do exactly what the original argument does, except only with rationals.

My understanding is that the argument is trying to show that all n cannot list all the reals. In my argument, It is the same. I want to show that all n cannot list all the rationals. (I am not actually trying to show this because I am certain that I have not disproved Cantor's proof. I only want to know why my "argument" does not work so that I can understand Cantor's argument better.)

 

[Mates]

Still confused: if you are excluding 3's how do you get 0.3333... ?

In the diagonal argument as I understand it, the number that is produced is the number that the natural numbers cannot list. My argument seems - to me - to show the same thing except only with rationals. But my "argument" cannot be true since we know that the set of rationals is not larger than the set of natural numbers. There is something about my argument that is wrong, but I still do not know what it is.

 

[NotJeffM]

The way I understand it is that in the argument (from Wikipedia), in base 2 for example, the rule is essentially either "not 0" or "not 1" at every diagonal position. With my example, I am just excluding 3 of the 9 other choices.

You were told early on that understanding in terms of bit strings may be more intuitive than understanding in terms of infinite expansions of numbers. But I assure you that jumping around from decimal expansions to binary expansions to bit strings is not the way to understand a subtle argument.

First, I have said repeatedly that you can use the diagonalization process to construct a real number that is NOT in the list of rational numbers. Please say you noticed that point.

Second, you have somehow mixed up the CORRECT idea that you can use the diagonalization process to construct some unknown number that is not in a list with the WRONG idea that you can use that process to construct a SPECIFIC number that is not in the list. You have chosen as your example a process that you believe will give you 1/3 out of a list that contains every rational number and therefore includes 1/3. It is obviously impossible to construct the decimal expansion of 1/3 from a list that contains the decimal expansion of 1/3 when the diagonalization process guarantees that the constructed number will differ from the decimal expansion of 1/3 in at least one digit. You can create through the diagonalization process on the compkete list of rational numbers a real number that is not in the list of rational numbers. All that means is that you have created a real number that is not rational and not in the list. Thus, the constructed number is irrational. There is no contradiction with the proof that you can create (in principle) a complete list of the rational numbers. We have a list of boy students. We identify a student who is a girl. That does invalidate the list of boy students.

Third, the whole chain of logic depends on being able to demonstrate a way to associate (in principle) each rational number with a unique natural number. I did that in an earlier post. With respect to the real numbers, we assume without proof that somehow we can construct a list of ALL real numbers without any demonstration. Then we use the diagonalization process to construct a real number that is not in the purported list of all real numbers. Thus, there is no way to construct such a list.

You have been right all along that there is a relevant difference between the rational numbers and the real numbers. In the case of the rational numbers, we can construct from a complete list of rational numbers a real number that is irrational. So what? In the case of the real numbers, if we can construct a complete list of the real numbers, we can then construct an unlisted real number. Disaster.

I have worked very hard on this. I am not willing to cover ground again that has been explained now in several posts. I am willing to answer questions about this post, but I no longer want to explain why you cannot take a list that contains 1/3 and construct 1/3 from that list and say 1/3 was not in the list to begin with. It is very easy, I admit, to get confused about the details, but the basic concept is simple. We can prove that in principle a list of all the rational numbers is possible. Therefore the fact that we can construct a real number from that list that is not in the list simply means that the constructed real number is not rational. But if we can construct a list of all the real numbers, we can then construct a real number that is not a real number because it is not in the list of all real numbers.

I am tired now and am going to have a whiskey and go to bed,

 

[blamocur]

In the diagonal argument as I understand it, the number that is produced is the number that the natural numbers cannot list. My argument seems - to me - to show the same thing except only with rationals. But my "argument" cannot be true since we know that the set of rationals is not larger than the set of natural numbers. There is something about my argument that is wrong, but I still do not know what it is.

In the diagonal argument you would not get 0.3333..., but a sequence without any 3's in it.

 

[JeffJo]

Cantor's Diagonal Argument is almost always taught incorrectly. It is very close to correct, but includes several picky little differences that not only confuse students who first learn them, but can be considered to make it wrong. (Don't worry, the correct proof is right.)

I'm hoping that this has been overlooked only because, as it was my first post in this forum, there was a delay in getting it released. I include it here so you can go read it.

It is important to realize that Cantor did not hypothesize a way to put the rationals and natural numbers into 1-1 correspondence. He specified conceptually how to do it. .... He assumes (for purposes of contradiction) that it is possible

No, it is vital to realize that he never, ever hypothesized a way to put the set he was working with into a 1-1 correlation with the natural numbers. He only worked with subsets that could. When it works, what diagonalization proves is that such a subset can't be the whole set.

Proof-by-contradiction is a tricky process, that often gets misrepresented. And CDA is almost always taught as an invalid example of it. Here's how it should go (and sorry, I don't know how to use math symbols).

1. You want to prove proposition A.
2. So you suppose that not(A) is true. (I don't like to say "assume" here, because you aren't claiming that it is, just exploring the consequences if it were.)
3. Prove two contradictory statements:
   1. not(A) --> B
   2. not(A) --> not(B)
4. Conclude that you did something wrong. If the two proofs are valid, that can only be that the unsupported supposition that not(A) is true. So A must be true.

The problem with how CDA is taught, is that it does not follow this form. What is does is more like:

1. You want to prove that, if S is a set of real numbers in [0,1], that not(A and B) is true.
2. Assume that (A and B) is true
3. Prove two contradictory statements:
   1. (A and B) --> A
   2. (A and B) --> not(A)
4. Conclude that not(A and B) is true.

In the usual presentation of CDA, A="Set S contains all the real numbers in [0,1]" and B="Set S can be put in a 1-1 correlation with the set N of all natural numbers."

And that creates a problem. The usual teaching never proves that (A and B)-->not(A), it proves B-->not(A). That is, it never uses the part of the assumption that every real number is in the list. The result is that the contradiction proves nothing, via proof-by-contradiction, about A. And the other "side" of the contradiction was an assumption, not something it actually proved. The conclusion of a proof-by-contradiction is invalid.

But that doesn't matter. All we want to prove is that A and B can't be true together. The fact that B-->not(A) does this, but to logically conclude that (A and B) is impossible requires an additional step. You could say that, by contraposition, B-->not(A) implies A-->not(B), proving that A and B can't be true together. Or, as Cantor did, show that:

1. (A and B)-->there is a real number r in [0,1] that is not in S.
2. (A and B)-->since r is in [0,1], r is in S.

+++++
I haven't looked hard at this disagreement over 1/3=0.33333... . But if that number is at position n in the list, then the nth digit of the "number" you create is not a 3. So you can't construct 1/3 from a list of all rational numbers. This is irrelevant, since no matter what set CDA applies to, you can't prove that the number you construct must always be rational. Even if it is possible to construct 1/3 from a list that does not include 1/3.
+++++
Notes: A 1-1 correspondence is technically a one-way relationship. It means that every member of the first set is matched to one member of the second set. A 1-1-1 correspondence works backwards, and is what Cantor called a 1-1 correlation.

And 3.14159265... is not the number "pi." It is a string of characters that represents the number pi in decimal notation. The distinction is important to CDA since 0.5000... and 0.4999... are difference strings, but represent the same number.

 

[Mates]

First, I have said repeatedly that you can use the diagonalization process to construct a real number that is NOT in the list of rational numbers. Please say you noticed that point.

[Before I start replying, I just want to say that I am going to directly respond to all of your feedback as best I can. Thanks for your patience (I hope I am not driving you to drink too much, lol)]

Yes I noticed that. I read everything as close as I can before I respond.

Second, you have somehow mixed up the CORRECT idea that you can use the diagonalization process to construct some unknown number that is not in a list with the WRONG idea that you can use that process to construct a SPECIFIC number that is not in the list.

Yes, that seems to be a difference. But I do not know why I can't use a specific rule.

You have chosen as your example a process that you believe will give you 1/3 out of a list that contains every rational number and therefore includes 1/3. It is obviously impossible to construct the decimal expansion of 1/3 from a list that contains the decimal expansion of 1/3 when the diagonalization process guarantees that the constructed number will differ from the decimal expansion of 1/3 in at least one digit.

Your critique to my response to this will be important. I have not fully explained where my thinking is in regards to this issue you bring up.

The reason I do understand why this is a problem is because the original argument seems to do the exact same thing. It assumes a list of reals, that we know in the end will not contain all the reals, but it generates a real number anyway. In my mind, it seems that you would have to have the same issue as you do with my argument. For my argument, we assume a list of rationals that we know in the end will not contain all the rationals, and then we generate a rational number anyway.

So that is my response to that issue.

You can create through the diagonalization process on the compkete list of rational numbers a real number that is not in the list of rational numbers. All that means is that you have created a real number that is not rational and not in the list. Thus, the constructed number is irrational.

I understand that, but like I allude to above, I do not believe that is what I am doing.

Third, the whole chain of logic depends on being able to demonstrate a way to associate (in principle) each rational number with a unique natural number. I did that in an earlier post. With respect to the real numbers, we assume without proof that somehow we can construct a list of ALL real numbers without any demonstration. Then we use the diagonalization process to construct a real number that is not in the purported list of all real numbers. Thus, there is no way to construct such a list.

Ok good, that is how I understand the diagonal argument as well.

You have been right all along that there is a relevant difference between the rational numbers and the real numbers. In the case of the rational numbers, we can construct from a complete list of rational numbers a real number that is irrational. So what? In the case of the real numbers, if we can construct a complete list of the real numbers, we can then construct an unlisted real number. Disaster.

That is not what I intend to do. If it is an irrational number that comes out, then I am completely lost, because I have no idea how that would happen given the instructions of my argument. Having said that, if we choose some other rule/s other than "not 3", in order to create an irrational number. And if it generates an irrational number, then it would not conflict with my current understanding of this topic.

I have worked very hard on this. I am not willing to cover ground again that has been explained now in several posts. I am willing to answer questions about this post, but I no longer want to explain why you cannot take a list that contains 1/3 and construct 1/3 from that list and say 1/3 was not in the list to begin with.

I hope that my response above will help us gain ground on that part of the topic.

 

[Mates]

but a sequence without any 3's in it.

Yes, that is what I want.

 

[NotJeffM]

I'm hoping that this has been overlooked only because, as it was my first post in this forum, there was a delay in getting it released. I include it here so you can go read it.

@JeffJo
I did not overlook it. I actually agree that the bit-string approach is a more intuitive way to grasp diagonalization. However, the OP seems to be interested in why diagonalization has different implications for rational numbers and for real numbers and has approached the problem through infinitely expanded decimal representations. I chose to stick with the approach that the OP was using. That does not mean I disagree with you that the bit-string approach is intuitive, but then you need further steps to move into what it means for the cardinality of different sets of numbers.

No, it is vital to realize that he never, ever hypothesized a way to put the set he was working with into a 1-1 correlation with the natural numbers. He only worked with subsets that could. When it works, what diagonalization proves is that such a subset can't be the whole set.

I think we do disagree here. I believe that Cantor did try to prove that the rational numbers were a denumerable set. My recollection (which may be incorrect) is that his proof had a minor flaw, which has since been corrected. I gave a demonstration that the cardinality of the rational numbers is less than or equal to the cardinality of the natural numbers. I conceded that to show equal cardinalities between the rational and natural numbers requires a further step, but that step is trivial. What the OP is asking about is why diagonalization has different consequences for rational and real numbers. In that context, a key difference is that there is a proof (not a mere hypothesis) that the rational numbers are denumerable.

Proof-by-contradiction is a tricky process, that often gets misrepresented. And CDA is almost always taught as an invalid example of it. Here's how it should go (and sorry, I don't know how to use math symbols).

1. You want to prove proposition A.
2. So you suppose that not(A) is true. (I don't like to say "assume" here, because you aren't claiming that it is, just exploring the consequences if it were.)
3. Prove two contradictory statements:
   1. not(A) --> B
   2. not(A) --> not(B)
4. Conclude that you did something wrong. If the two proofs are valid, that can only be that the unsupported supposition that not(A) is true. So A must be true.

The problem with how CDA is taught, is that it does not follow this form. What is does is more like:

1. You want to prove that, if S is a set of real numbers in [0,1], that not(A and B) is true.
2. Assume that (A and B) is true
3. Prove two contradictory statements:
   1. (A and B) --> A
   2. (A and B) --> not(A)
4. Conclude that not(A and B) is true.

In the usual presentation of CDA, A="Set S contains all the real numbers in [0,1]" and B="Set S can be put in a 1-1 correlation with the set N of all natural numbers."

And that creates a problem. The usual teaching never proves that (A and B)-->not(A), it proves B-->not(A). That is, it never uses the part of the assumption that every real number is in the list. The result is that the contradiction proves nothing, via proof-by-contradiction, about A. And the other "side" of the contradiction was an assumption, not something it actually proved. The conclusion of a proof-by-contradiction is invalid.

But that doesn't matter. All we want to prove is that A and B can't be true together. The fact that B-->not(A) does this, but to logically conclude that (A and B) is impossible requires an additional step. You could say that, by contraposition, B-->not(A) implies A-->not(B), proving that A and B can't be true together. Or, as Cantor did, show that:

1. (A and B)-->there is a real number r in [0,1] that is not in S.
2. (A and B)-->since r is in [0,1], r is in S.

I am truly not sure if you are criticizing the substance of what I have been saying or my manner of expressing that substance I or how other people present their arguments. In any case my argument is

Assume there is a list (denumerable set) of unique decimal representations of each real number in the interval (0,1).

Construct the representation r via diagonalization.

r is a representation of a real number in the interval (0, 1)

By our assumption r is in the list.

By construction r is not in the list.

Therefore there is no list (denumerable set) of unique decimal representations of each real number in the interval (0, 1).

Please explain the flaw in that argument. (I concede that we need a further step to jump from representations of real numbers to the real numbers themselves, but that extra step is trivial).

 

[Mates]

And 3.14159265... is not the number "pi." It is a string of characters that represents the number pi in decimal notation.

Interesting, but I do not understand what you mean. What is the difference?

 

[JeffJo]

I did not overlook it. I actually agree that the bit-string approach is a more intuitive way to grasp diagonalization.

My main point was that CDA is usually taught incorrectly. The set used is just one example, that I included to prove I knew whaty I was talking about. But the correct proof answers the OP's questions. Simplified>

1. Let T be the set of interest.
2. Let S be any subset of T that actually can be enumerated.
3. Use diagonalization to construct an element s that:
   1. You can prove is in T.
   2. You know is not in S because it is different from each sn.
4. This proves that any set S that can actually be enumerated is missing at least one constructable element s.
5. Assume T can be enumerated.
   1. Apply #3 to find an s that is not in the enumerated set, T.
   2. Apply the definition of "all" to find that this s in T.
   3. Contradiction disproves the assumption if #5.

CDA can't be applied to the rational numbers because you can't prove #3.1 in general.

However, the OP seems to be interested in why diagonalization has different implications for rational numbers and for real numbers

Because for real numbers, once you resolve the duplicate-representation issue, you can prove #3.1 in general.

I think we do disagree here. I believe that Cantor did try to prove that the rational numbers were a denumerable set.

Yes, he did, with a similar approach. But not the same one. Intuitively, you can trace your finger through an table of rational numbers, with the numerator labeling the columns and the denominator labeling the rows. Whether or not you remove duplicates (like 2/4=1/2), you will hit every fraction.

My recollection (which may be incorrect) is that his proof had a minor flaw, which has since been corrected.

His first proof, which did use real numbers in any interval [a,b] did have a flaw, that he claims to have corrected. It is above my pay grade. This proof referenced that one, and said it was simpler and did not depend on the properties of irrational numbers.

I am truly not sure if you are criticizing the substance of what I have been saying or my manner of expressing that substance I or how other people present their arguments. In any case my argument is

I'm not criticizing either. I'm trying to correct how CDA is understood. One such misunderstanding is the word "each" in the following:

Assume there is a list (denumerable set) of unique decimal representations of each real number in the interval (0,1).

This is not a part of CDA. Putting it in leads to endless cycles of misplacing where someone understands it incorrectly. The subtle part is that you can't "Construct the representation r via diagonalization" unless you know the list exists. But you can apply the result in my step #4 if you assume a list could exist. I know this is extremely finicky, but many students balk at assuming you actually have done what you afre trying to prove can't be done.

And the reason I mention it here, is because the diaonalization process never has to be applied to a list of rational numbers that includes 1/3. You could, theoretically, create a highly selective list that does not include 1/3 and produces 1/3 as the diagonal. But you can't prove the result is a rational number when you use any list that is denumerable.

 

[JeffJo]

Interesting, but I do not understand what you mean. What is the difference?

Remember "the artist formally known as Prince"? He legally changed his name to an unpronounceable symbol. That symbol wasn't the man himself, it was just a name. When he changed his name back to "Prince," those letters were not the man himself, they were just a name.

My point is that "3.14159265..." is one way we name the number pi. It stands for the formula 3+1/10+4/100+... . CDA works on that name, not the number itself. And the reason for the point is that CDA never concerns itself with how you might interpret the strings it uses.

 

[NotJeffM]

[Before I start replying, I just want to say that I am going to directly respond to all of your feedback as best I can. Thanks for your patience (I hope I am not driving you to drink too much, lol)]

I’m married and have children so you won’t be the one that drives me to drink.

Yes, that seems to be a difference. But I do not know why I can't use a specific rule.

Now we may be getting somewhere. You are correct that the diagonalization process is not very constraining. It says replace every indicated digit with a different digit other than 9. It gives you eight choices. You could specify any rule of replacement that you want provided it meets those general constraints. So you CAN use a more specific rule.

What I said is different. I said that you cannot devise a diagonalization rule that guarantees a specific number will be the result. ”Aha,” I hear you say, “the specific rule is that each indicated digit is to be replaced by 3. That will let me create 0.3333…..” That is true ONLY IF the list of numbers being diagonalized consists of numbers that do not contain a 3 anywhere in their infinite decimal expansion. So, to be precise, I should have said that you cannot devise a specific diagonaliztion rule that will generate a specific number from the lists of numbers that we are interested in.

In particular, you cannot use any diagonalization rule on the list of all rational numbers to create the decimal expansion of 1/3 because that list contains the decimal expansion of 1/3 and so the number created must differ from 1/3. The whole point is to create a number that, by construction, cannot be in the list.

Your critique to my response to this will be important. I have not fully explained where my thinking is in regards to this issue you bring up.

The reason I do understand why this is a problem is because the original argument seems to do the exact same thing. It assumes a list of reals, that we know in the end will not contain all the reals, but it generates a real number anyway. In my mind, it seems that you would have to have the same issue as you do with my argument. For my argument, we assume a list of rationals that we know in the end will not contain all the rationals, and then we generate a rational number anyway.

Yes, here we get to the flaw in your reasoning. We do not simply “assume” some list of rational numbers “that we know in the end will not contain all the rational numbers.” Obviously if we know in advance that the list does not contain all the rational numbers, there is no contradiction in finding a rational number that is not listed. The only relevant list is the list of all rational numbers. Nor do we assume that such a list is possible (in principle). We have previously proved that it is possible. We now apply diagonalization to get a real number that is not listed. Therefore that number cannot be a rational number. It must be an irrational number. There is no contradiction. A girl student not in the list of all boy students does not challenge the validity of that list.

With the real numbers, the circumstances differ in two respects. First, we have not previously proved that a list of even the real numbers in (0, 1) is possible. We say suppose it is possible. We are not saying we know in advance that it is not possible. Just the opposite. We now apply the diagonalization process to our list to create a real number that is not listed among all the real numbers. A student not in the list of all students destroys the validity of that list.

Diagonalization creates a real number not in the list. With respect to rational numbers, that is ok because a real number may be rational or not rational (irrational). We never said irrational numbers were in a list of rational number. With respect to real numbers, it is not ok. A real number not in the list is either rational or irrational, but in either case it was supposed to be in the list and isn’t.

The irrational numbers provide an escape hatch for the rational numbers, but not for the real numbers.

Let’s pause here because I truly do not know where we go after this point.

 

[JeffJo]

I am absolutely under no illusion that I have proved anything. I brought this up because it means that I do not understand something about the Cantor's diagonal argument.

Here is what it looks like. Just remember that the rule is "not 3" for the "intersection" of the nth row to the nth column.View attachment 35292

This is why I ignored this. My understanding is that the rule applies to the construction of the enumeration S1, S2, S3, .... Not to the creation of the "new number" S from what is an arbitrary enumeration. You need two rules for that. In this example they are:

• If Sn(n) is not 3, then S(n)=3.
• If Sn(n)=3, then S(n)=????? . You don't specify.

The problem is that you don't get for control the Sn. Some digits that you apply a diagonalizing rule to will be 3s - for example, if there is an Sn that already is 1/3.

But this still doesn't make sense. So maybe of you explain what "the rule is not 3" means? What does it apply to?

 

[Mates]

I hear you say, “the specific rule is that each indicated digit is to be replaced by 3. That will let me create 0.3333…..” That is true ONLY IF the list of numbers being diagonalized consists of numbers that do not contain a 3 anywhere in their infinite decimal expansion.

Why can't the list of numbers contain a 3 anywhere? The rule is only for one decimal place, specifically the nth decimal place for the nth row. I must be missing something here.

Yes, here we get to the flaw in your reasoning. We do not simply “assume” some list of rational numbers “that we know in the end will not contain all the rational numbers.” Obviously if we know in advance that the list does not contain all the rational numbers, there is no contradiction in finding a rational number that is not listed. The only relevant list is the list of all rational numbers. Nor do we assume that such a list is possible (in principle). We have previously proved that it is possible. We now apply diagonalization to get a real number that is not listed. Therefore that number cannot be a rational number. It must be an irrational number. There is no contradiction. A girl student not in the list of all boy students does not challenge the validity of that list.

I should've been more clear. I meant that you and I know that the list we will make for the reals will not contain all the reals because we already know the outcome of the argument.

With the real numbers, the circumstances differ in two respects. First, we have not previously proved that a list of even the real numbers in (0, 1) is possible. We say suppose it is possible. We are not saying we know in advance that it is not possible. Just the opposite. We now apply the diagonalization process to our list to create a real number that is not listed among all the real numbers. A student not in the list of all students destroys the validity of that list.

Okay, but I do not see how that difference is a problem for my argument. I am trying to figure out exactly how my argument fails.

Diagonalization creates a real number not in the list. With respect to rational numbers, that is ok because a real number may be rational or not rational (irrational). We never said irrational numbers were in a list of rational number. With respect to real numbers, it is not ok. A real number not in the list is either rational or irrational, but in either case it was supposed to be in the list and isn’t.

The irrational numbers provide an escape hatch for the rational numbers, but not for the real numbers.

I understand that. I know that we can generate an irrational number using the argument. But that is not what I want to explore. I only want to explore what part of my argument fails when generating a rational number. The point is to attempt to parallel Cantor's argument and then to find out how it fails.

Let’s pause here because I truly do not know where we go after this point.

I am thinking that the solution is probably really obvious, but now we are very deep and our minds are looking for something very subtle.

 

[NotJeffM]

Why can't the list of numbers contain a 3 anywhere? The rule is only for one decimal place, specifically the nth decimal place for the nth row. I must be missing something here.



I should've been more clear. I meant that you and I know that the list we will make for the reals will not contain all the reals because we already know the outcome of the argument.



Okay, but I do not see how that difference is a problem for my argument. I am trying to figure out exactly how my argument fails.



I understand that. I know that we can generate an irrational number using the argument. But that is not what I want to explore. I only want to explore what part of my argument fails when generating a rational number. The point is to attempt to parallel Cantor's argument and then to find out how it fails.



I am thinking that the solution is probably really obvious, but now we are very deep and our minds are looking for something very subtle.

The problem is that I have no clue what your argument is. It seems to wander all over the place.

To the extent that your argument is that by some more specific rule, you can create 1/3 through diaganolization of the list of all rational numbers, it is necessarily true that diagonalization creates a real number that is not in the given list and that 1/3 is in the given list so there is no specific rule that can create 1/3.

If your argument is that you can create some unknown rational number by diagonalizing the list of all rational numbers, that argument fails for the exact same reason. The only numbers you can construct by diagonalizing the rational numbers are irrational numbers. That is what distinguishes the implication of applying diaganolization to the rational and to the real numbers. Diaganolization creates a real number that is not in the list. That is not a problem if the list is of rational numbers. Nor is it a problem if the list contains only some real numbers but not all. It creates a huge problem for any claim that the list contains all real numbers.