problem with cantor diagonal argument

[KJW]

1 hour ago, phyti said:

Does the progression of finite lists lead to transfinite lists?

The list of sequences is infinite, specifically countably infinite because it is a list. It is only displayed as finite because of the limitations of illustrations. In fact, Cantor proved using set-theoretical notation that a set and the set of all its subsets (its power set) cannot be placed into one-to-one correspondence. The diagonal argument is a mere illustration of the proof for a countably infinite set. What Cantor proved is that there is an infinite sequence of distinct infinite cardinalities: natural numbers, power set of natural numbers, power set of power set of natural numbers, power set of power set of power set of natural numbers, etc.

 

 

1 hour ago, phyti said:

Cantor's Illusion.pdfCantor's Illusion.pdf 109.75 kB · 1 download

I'm not opening that. Please present it in a form that does not require opening a file.

 

 

[phyti]

KJW;

'

The list of sequences is infinite, specifically countably infinite because it is a list. It is only displayed as finite because of the limitations of illustrations. In fact, Cantor proved using set-theoretical notation that a set and the set of all its subsets (its power set) cannot be placed into one-to-one correspondence. The diagonal argument is a mere illustration of the proof for a countably infinite set. What Cantor proved is that there is an infinite sequence of distinct infinite cardinalities: natural numbers, power set of natural numbers, power set of power set of natural numbers, power set of power set of power set of natural numbers, etc.

...

I'm not opening that. Please present it in a form that does not require opening a file.

The typical answer, repeat Cantor's argument using set theory math babble, and assume there are no errors.We know there is no literal list without an end, which is a contradiction of terms. We live in a world of finite objects with boundaries which enables measurement. You can't measure length of a rod with only one end. The use of finite lists is to determine their form as the number of sequences increases, augmented with permutation methods.The 1891 paper uses symbols other than numbers and is not about the power set or any form of number.

The pdf format is an efficient form that anyone with the 'reader' can access. Mine is 4 pages. Most repositories scan their files before downloading, and you should have that same option with antivirus software.

 

Edited December 11, 2023 by phyti

[KJW]

4 hours ago, phyti said:

The typical answer, repeat Cantor's argument using set theory math babble, and assume there are no errors.

No, anyone who understands the proof can see that there are no errors. It's not a difficult proof, one can work through the proof to deduce that it is correct without assuming that it is correct.

Calling it "set theory math babble" doesn't bode well for your credibility on the subject.

 

 

4 hours ago, phyti said:

We know there is no literal list without an end, which is a contradiction of terms. We live in a world of finite objects with boundaries which enables measurement. You can't measure length of a rod with only one end.

We are talking about the mathematical realm, not the physical realm. You cannot use physical reality to argue anything about mathematics. Doing so also doesn't bode well for your credibility on the subject. I think it is fair to say that in mathematics, everything exists unless it can be mathematically proven not to exist. In the case of infinite sets, we know they mathematically exist because there is no largest natural number, and therefore we have an example of an infinite set, the natural numbers.

 

 

[phyti]

KJW;

Theories are proposed but only accepted with evidence of experience agreeing with prediction. Mathematical conjectures are no different. Other conjectures have been proven and disproved. Thus I can't agree with accepting math as true without proof no more than some one guilty of a crime without evidence.

I have included the paper in png format so you can avoid any risk.

The issue is geometric form of the list, a common early objection to cda.
His 'contradiction' is essentially a tautology equivalent to 'a thing can't be in two different locations at the same time'.

His flawed vision of an infinite list easily qualifies as a contradiction.

 

[KJW]

Looking at your paper, it seems to me that you misunderstand the theorem being proven by Cantor. Cantor is proving that there does not exist a one-to-one correspondence between a set and the set of all the subsets of that set. It is true for finite sets in a rather trivial manner, and it is for infinite sets that the theorem is truly interesting. But let's consider the case of the finite set {1,2,3,4,5,6,7,8}. Vertically on the left is a list of the eight elements, and to the right of each of the elements is a representation of a subset of that set. If the subset contains a particular element, that element will be shown in its numerical position in the sequence. Otherwise, the numerical position will contain "o":

1=1o345o78
2=1o3o56o8
3=1o3o5o78
4=12oo5678
5=o234o678
6=1o34ooo8
7=12oo567o
8=1oo4oo78

Each subset is unique in the list. Now consider the diagonal:

1o3ooo78

The complement of the diagonal:

o2o456oo

differs from the first subset in the first element, differs from the second subset in the second element, differs from the third subset in the third element, ... , differs from the eighth subset in the eighth element, and so differs from every subset in the list. The complement is a subset of the eight elements but is not one of the eight subsets in the list. Furthermore, every possible list of eight subsets will have a complement of the diagonal that is missing from that list, proving that there does not exist a one-to-one correspondence between a set and the set of all the subsets of that set.

 

 

Edited December 13, 2023 by KJW

[phyti]

KJW;

Cantor is proving that there does not exist a one-to-one correspondence between a set and the set of all the subsets of that set.

That is the power set and not the subject of the 1891 paper analyzed.

From the translation:

[defines set]

"Let M be the totality [Gesamtheit] of all elements E."

[defines elements]

"EI = (m, m, m, m, … ),

EII = (w, w, w, w, … ),

EIII = (m, w, m, w, … )."

[states cardinality of M is different from N]

"I maintain now that such a manifold [Mannigfaltigkeit] M does not have the power of the series 1, 2, 3, …, v, …."

[displays M in terms of coordinates (u, v)

"For proof, let there be

E1 = (a1.1, a1.2, … , a1,v, …)

E2 = (a2.1, a2.2, … , a2,v, …)

Eu = (au.1, au.2, … , au,v, …)"

[defines diagonal b, forms negation as E0]

"Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v."

He doesn't consider any relation of u to v, which is s = 2v in this case.
If the sequences involved 'how many unique patterns can be formed with 5 unique symbols, the relation would be s=5!, i.e. the geometric form depends on the rule of formation.
As shown in fig.3, the red diagonal sequence of 64 columns is impossible in a list of sequences 6 characters long, which is true for all v which =n from set N.

[KJW]

14 hours ago, phyti said:

19 hours ago, KJW said:

Cantor is proving that there does not exist a one-to-one correspondence between a set and the set of all the subsets of that set.

That is the power set and not the subject of the 1891 paper analyzed.

According to the Wikipedia article "Cantor's theorem", that is precisely what he is proving:

Quote

In mathematical set theory, Cantor's theorem is a fundamental result which states that, for any set A, the set of all subsets of A, the power set of A, has a strictly greater cardinality than A itself.

I should point out that while I am willing to debate the principles associated with Cantor's theorem, including the diagonal argument, I am unwilling to debate over a translation of the precise wording of an 1891 paper. You presented a paper in which it appears that you have a problem with the relationship between the length of the list and the length of the sequences, indicating that the diagonal argument requires that these two lengths be equal. In the post that you quoted, I showed you why they are equal, but you didn't respond to this. Quite simply, the length of the list is the number of elements in the set, and the length of the sequences is also the number of elements in the set because a subset of a set is made up of elements in the set, and the representation of the subset is a sequence indicating whether or not each element in the set is also in the subset.

 

Edited December 14, 2023 by KJW

[phyti]

KJW;

Don't just repeat what you read.

Think about it as if you were doing research.
In your 8 column example and all square lists, the diagonal (b) and its complement (not b) cannot coexist in the same square space by definition. That is not a revelation from above.
If v=8 columns and the set of symbols is {0 to 8}, then number of sequences/patterns is s=98 = 43,046,721.
(Not b) can exist in the missing portion of the list. The list is incomplete!

 

[KJW]

1 hour ago, phyti said:

Don't just repeat what you read.

Is that what you think I'm doing? No, I am trying point out what I believe your error is. If that seems like I'm repeating what I read, then it would seem that you are not understanding my point.
 

 

1 hour ago, phyti said:

Think about it as if you were doing research.

I'm not sure what you mean by this. The objective for me is clear: to explain Cantor's theorem and its proof. There is no doubt in my mind about these, but trying to explain them to you is proving to be a challenge. It's not clear to me that you actually know what it is that is being proven.
 

 

1 hour ago, phyti said:

In your 8 column example and all square lists, the diagonal (b) and its complement (not b) cannot coexist in the same square space by definition. That is not a revelation from above.
If v=8 columns and the set of symbols is {0 to 8}, then number of sequences/patterns is s=98 = 43,046,721.
(Not b) can exist in the missing portion of the list. The list is incomplete!

 

It appears to me that you have misunderstood my 8 x 8 array. I was hoping that I had explained it, but apparently not well enough. The set of symbols is not {o,1,2,3,4,5,6,7,8}, but simply the binary "this element is in the subset" / "this element is not in the subset" choice. That is, there are not 98 = 43,046,721 possible sequences, but only 28 = 256. In other words, in my 8 x 8 array, you could replace the numbers "1", "2", "3", "4", "5", "6", "7", "8" with "Y" and "o" with "N". I apologise for the confusion.

 

You appear to be treating Cantor's theorem as being about lists and sequences. But it is better to treat Cantor's theorem in terms of a set, its elements, and its subsets. Cantor's theorem then says that for a given set, there are more subsets than elements. The 8 x 8 array is about a set with 8 elements. For this set, there are 256 subsets which is obviously greater than the 8 elements, and thus Cantor's theorem is proven for this set. The diagonal argument wasn't really needed to show this, but I did used it to show how it is applied. In the case where the set is the set of natural numbers, the diagonal argument becomes valuable.

 

You mention "missing sequences" as if their existence is a problem. No, their existence is what is being proven. It's obvious in the case of finite sets, but also true for infinite sets.

 

 

 

Edited December 16, 2023 by KJW

[phyti]

KJW;

1. You appear to be treating Cantor's theorem as being about lists and sequences. But it is better to treat Cantor's theorem in terms of a set, its elements, and its subsets. 

2. Cantor's theorem then says that for a given set, there are more subsets than elements. The 8 x 8 array is about a set with 8 elements. For this set, there are 256 subsets which is obviously greater than the 8 elements, and thus Cantor's theorem is proven for this set.

3. The diagonal argument wasn't really needed to show this, but I did used it to show how it is applied. In the case where the set is the set of natural numbers, the diagonal argument becomes valuable.

4. In other words, in my 8 x 8 array, you could replace the numbers "1", "2", "3", "4", "5", "6", "7", "8" with "Y" and "o" with "N". I apologise for the confusion.

5. You mention "missing sequences" as if their existence is a problem. No, their existence is what is being proven. It's obvious in the case of finite sets, but also true for infinite sets.

1. It's about Cantor's diagonal argument, not his theorem or the power set. The set is M as he defined it in his paper.

2. This was already known using any system of symbols for counting, long before Cantor.

3. Cantor envisioned a square list, which means the diagonal b intersected every row in the list, thus excluding the negation (not b). It's a self fulfilling prophecy. The problem, the row count u equals the column count v, contradicting u>v.

4. Cantor uses {m and w} alphabetical symbols as the building blocks for the E sequences/patterns. {H and T}, {0 and 1}, or any binary set of 2 symbols would work. The symbols are not composite things. It's not determining true or false regarding the patterns, they are given. The question, 'is the diagonal 'b' a new pattern, not in the list?'.

It is not new because his list is incomplete.

5. We prove they exist by calculation, while Cantor ignores them.

----------------------------------------------------------

Compare the list of integers in the base 10 system, to the list for Cantor's sequences.

Each position to the left has the capacity of 10 times the position to its right.

The number of positions increases linearly, the number of integers increases exponentially.

 

 

[KJW]

1 hour ago, phyti said:

It's about Cantor's diagonal argument, not his theorem or the power set.

How can it not be about his theorem or the power set? Firstly, Cantor's diagonal argument is about proving his theorem. It makes no sense at all to discuss the diagonal argument without mentioning his theorem. And secondly, his theorem is about the power set in relation to the set.

 

 

1 hour ago, phyti said:

Cantor envisioned a square list, which means the diagonal b intersected every row in the list, thus excluding the negation (not b). It's a self fulfilling prophecy.

That (not b) is not in the square list is the whole point of the diagonal argument. It shows that the number of possible sequences is greater than the number of rows in the square list. It's only "self-fulfilling" because for a finite list, the theorem is so obviously true. But it is much less obvious for an infinite list.

 

 

1 hour ago, phyti said:

The problem, the row count u equals the column count v, contradicting u>v.

No, the row count u is supposed to be equal to the column count v. The row count is not the number of possible sequences (the number of subsets of the given set). The row count is the number of elements of the given set (equal to the column count).

 

 

1 hour ago, phyti said:

The question, 'is the diagonal 'b' a new pattern, not in the list?'.

It is not new because his list is incomplete.

We prove they exist by calculation, while Cantor ignores them.

The question is "Is the negation of the diagonal in the square list?" The diagonal argument is that it is not. If you want to extend the list to cover all possible sequences, then it is only natural that the negation of the diagonal is in the extended part of the list. But the diagonal argument is about the "incomplete" square list only.

 

 

Edited December 17, 2023 by KJW

[phyti]

KJW;

1. How can it not be about his theorem or the power set? Firstly, Cantor's diagonal argument is about proving his theorem. It makes no sense at all to discuss the diagonal argument without mentioning his theorem. And secondly, his theorem is about the power set in relation to the set.

 

2. That (not b) is not in the square list is the whole point of the diagonal argument. It shows that the number of possible sequences is greater than the number of rows in the square list. It's only "self-fulfilling" because for a finite list, the theorem is so obviously true. But it is much less obvious for an infinite list

 

3. No, the row count u is supposed to be equal to the column count v. The row count is not the number of possible sequences (the number of subsets of the given set). The row count is the number of elements of the given set (equal to the column count).

 

4. If you want to extend the list to cover all possible sequences, then it is only natural that the negation of the diagonal is in the extended part of the list. But the diagonal argument is about the "incomplete" square list only.

 

1. The power set of S having greater cardinality than S is a subject of combinatorics, and was studied in early history. A typical example:

{ab} {ac} {bc} {a} {b} {c} are the subsets of {abc}. I ignore the redundant feature of proper vs.improper and the universal empty set {}. No basis for attributing it to Cantor. It does not involve the diagonal argument.

2. What is there to argue. If (not b) is the negation of b, simple logic/reasoning prohibits both being in the same space. Wherever they intersect can only be one symbol. My cell phone can't be in the house and not in the house simultaneously.

3. Check fig.3 in post Dec 12. Cantor's red diagonal has many more characters than the other sequences.

4. The red is exactly my argument. What follows has no value. A subset contains less than the complete set by definition.

 

[KJW]

10 hours ago, phyti said:

The power set of S having greater cardinality than S is a subject of combinatorics, and was studied in early history. A typical example:

{ab} {ac} {bc} {a} {b} {c} are the subsets of {abc}. I ignore the redundant feature of proper vs.improper and the universal empty set {}. No basis for attributing it to Cantor. It does not involve the diagonal argument.

I have already said that for a finite set, it is obvious that the number of subsets is greater than the number of elements, and that it doesn't require the diagonal argument to prove this. The reason I did apply the diagonal argument was to demonstrate the procedure of the diagonal argument, rather than to actually prove anything. But Cantor's theorem is not just about finite sets. Although Cantor's theorem is true for finite sets, including the empty set, it is for infinite sets that the theorem is intended to be applied. And it is for infinite sets that the diagonal argument demonstrates its power.

 

 

10 hours ago, phyti said:

A subset contains less than the complete set by definition.

No. Although it is true for finite sets that a proper subset has fewer elements than the set, it is not true for infinite sets. Indeed, it is a defining property of infinite sets that they can be mapped one-to-one onto proper subsets. As far back as Galileo Galilei, it was demonstrated that the number of square numbers is equal to the number of numbers themselves:

[math]1 \longleftrightarrow 1[/math]
[math]2 \longleftrightarrow 4[/math]
[math]3 \longleftrightarrow 9[/math]
[math]4 \longleftrightarrow 16[/math]
[math]5 \longleftrightarrow 25[/math]
[math]6 \longleftrightarrow 36[/math]
[math]7 \longleftrightarrow 49[/math]
[math]8 \longleftrightarrow 64[/math]
[math]...[/math]
[math]n \longleftrightarrow n^2[/math]
[math]...[/math]

 

Indeed, one can even form a one-to-one correspondence between [math]n[/math] and [math]2^n[/math] for all [math]n \in \textstyle \mathbb {N}[/math]:

[math]1 \longleftrightarrow 2[/math]
[math]2 \longleftrightarrow 4[/math]
[math]3 \longleftrightarrow 8[/math]
[math]4 \longleftrightarrow 16[/math]
[math]5 \longleftrightarrow 32[/math]
[math]6 \longleftrightarrow 64[/math]
[math]7 \longleftrightarrow 128[/math]
[math]8 \longleftrightarrow 256[/math]
[math]...[/math]
[math]n \longleftrightarrow 2^n[/math]
[math]...[/math]

This allows one to create the following one-to-one correspondence of the natural numbers to subsets of the natural numbers:

[math]1 \longleftrightarrow 00000000...[/math]
[math]2 \longleftrightarrow 10000000...[/math]
[math]3 \longleftrightarrow 01000000...[/math]
[math]4 \longleftrightarrow 11000000...[/math]
[math]5 \longleftrightarrow 00100000...[/math]
[math]6 \longleftrightarrow 10100000...[/math]
[math]7 \longleftrightarrow 01100000...[/math]
[math]8 \longleftrightarrow 11100000...[/math]
[math]...[/math]

Note that applying Cantor's diagonal argument to this gives [math]11111111...[/math] as the representation of the subset of the natural numbers that is not in this list. That is, it appears as if the set of the natural numbers itself is the "final" subset of the set of all subsets. But note that the above list is not a list of the set of all subsets but is a list of the set of all finite subsets, and this can be mapped one-to-one onto the set of natural numbers. The subset corresponding to [math]11111111...[/math], the set of natural numbers itself, is not a finite subset so it doesn't belong to the above list of subsets. In other words, Cantor's diagonal argument is only excluding a subset that would be excluded anyway.

 

 

Edited December 19, 2023 by KJW

[studiot]

1 hour ago, KJW said:

I have already said that for a finite set, it is obvious that the number of subsets is greater than the number of elements, and that it doesn't require the diagonal argument to prove this. The reason I did apply the diagonal argument was to demonstrate the procedure of the diagonal argument, rather than to actually prove anything. But Cantor's theorem is not just about finite sets. Although Cantor's theorem is true for finite sets, including the empty set, it is for infinite sets that the theorem is intended to be applied. And it is for infinite sets that the diagonal argument demonstrates its power.

 

 

No. Although it is true for finite sets that a proper subset has fewer elements than the set, it is not true for infinite sets. Indeed, it is a defining property of infinite sets that they can be mapped one-to-one onto proper subsets. As far back as Galileo Galilei, it was demonstrated that the number of square numbers is equal to the number of numbers themselves:

1⟷1
2⟷4
3⟷9
4⟷16
5⟷25
6⟷36
7⟷49
8⟷64
...
n⟷n2
...

 

Indeed, one can even form a one-to-one correspondence between n and 2n for all n∈N :

1⟷2
2⟷4
3⟷8
4⟷16
5⟷32
6⟷64
7⟷128
8⟷256
...
n⟷2n
...

This allows one to create the following one-to-one correspondence of the natural numbers to subsets of the natural numbers:

1⟷00000000...
2⟷10000000...
3⟷01000000...
4⟷11000000...
5⟷00100000...
6⟷10100000...
7⟷01100000...
8⟷11100000...
...

Note that applying Cantor's diagonal argument to this gives 11111111... as the representation of the subset of the natural numbers that is not in this list. That is, it appears as if the set of the natural numbers itself is the "final" subset of the set of all subsets. But note that the above list is not a list of the set of all subsets but is a list of the set of all finite subsets, and this can be mapped one-to-one onto the set of natural numbers. The subset corresponding to 11111111... , the set of natural numbers itself, is not a finite subset so it doesn't belong to the above list of subsets. In other words, Cantor's diagonal argument is only excluding a subset that would be excluded anyway.

 

 

Excellent summary. +1

[Genady]

The "diagonal argument" can be expressed purely algebraically, without lists, rows, diagonals, and any other "visual aids". I think that such algebraic presentation would eliminate a lot of confusion.

Edited December 19, 2023 by Genady

[KJW]

1 hour ago, Genady said:

The "diagonal argument" can be expressed purely algebraically, without lists, rows, diagonals, and any other "visual aids". I think that such algebraic presentation would eliminate a lot of confusion.

Another advantage of such algebraic presentation is that it can be applied to uncountably infinite sets.

 

 

[phyti]

KJW;

 

In his 1891 paper Cantor defined the elements of M as horizontal sequences of m and w.

"Let M be the totality [Gesamtheit] of all elements E."

Here is the beginning of the list corresponding to M with {0, 1} substituted for {m. w}.

 

1. 000000...

2. 100000...

3. 110000...

4. 111000...

5. 111100...

6. 111110...

∶

'Infinity' is not a number, It's a state/property of not having a boundary, thus can't be measured (counted). There are no degrees of infinite. A thing has a boundary or it doesn't. If you know the value of א tell us what it is.

In choosing an integer in N as a corresponding identity for a sequence, the list automatically becomes infinite with no last sequence. There is no last integer. For every sequence added to the list, there is another integer n+1. The set N is inexhaustible.

The blue symbols represent a different definition of a sequence b that extends in two dimensions, u and v. Each of those symbols is contained in a different E.

Cantor never proved b was a new sequence. Notice b is the same as E1.

 

[KJW]

18 hours ago, phyti said:

Cantor never proved b was a new sequence. Notice b is the same as E1.

I don't know what point you are trying to make with this. The diagonal sequence is only used to form the negation of the diagonal sequence, and it is the negation of the diagonal sequence that is not in the list of sequences. Cantor's diagonal argument says nothing about the diagonal sequence itself, and there is no reason why it can't be in the list of sequences. Whether the diagonal sequence is or isn't in the list of sequences is not important to the diagonal argument.

 

 

18 hours ago, phyti said:

'Infinity' is not a number

Nobody said it was a number.

 

 

18 hours ago, phyti said:

It's a state/property of not having a boundary

...

A thing has a boundary or it doesn't.

Although I agree about infinity being unbounded, are you sure that is the only property of infinity?

 

 

18 hours ago, phyti said:

There are no degrees of infinite.

Cantor's theorem says otherwise. Indeed, Cantor's theorem implies that there are at least ℵ0 different magnitudes of infinity.

 

 

18 hours ago, phyti said:

If you know the value of א tell us what it is.

ℵ0 is the cardinality of the natural numbers. Perhaps you could tell me what the value of zero is.

 

 

18 hours ago, phyti said:

There is no last integer... The set N is inexhaustible.

Congratulations on proving that N is infinite! But even though N is "inexhaustible", there still isn't enough elements in it to cover the set of all subsets of N .

We have N as the reference set of cardinality ℵ0 . We can determine if other sets have a cardinality of ℵ0 by determining if it's possible to place their elements into one-to-one correspondence with the elements of N . This may require some ingenuity. For example, one can't simply list the rational numbers Q in numerical order. But one can list them in the order:

 

proving that [math]\textstyle \mathbb {Q}[/math] has the same cardinality as [math]\textstyle \mathbb {N}[/math]. But in the case of the real numbers [math]\textstyle \mathbb {R}[/math], the diagonal argument proves that there does not exist any such list, proving that the cardinality of [math]\textstyle \mathbb {R}[/math] is greater than the cardinality of [math]\textstyle \mathbb {N}[/math]. That is, if one can find a one-to-one correspondence between a given set and [math]\textstyle \mathbb {N}[/math], this proves that the cardinality of the set is [math]\aleph_0[/math], whereas if one can prove that no such one-to-one correspondence exists, then this proves that the cardinality of the set is not [math]\aleph_0[/math].

 

 

Edited December 21, 2023 by KJW

[studiot]

18 hours ago, phyti said:

'Infinity' is not a number, It's a state/property of not having a boundary, thus can't be measured (counted). There are no degrees of infinite. A thing has a boundary or it doesn't.

 

1 hour ago, KJW said:

Although I agree about infinity being unbounded, are you sure that is the only property of infinity?

 

An Infinity can be bounded below or bounded above yet unbounded in the other.

[phyti]

KJW;

I don't know what point you are trying to make with this. The diagonal sequence is only used to form the negation of the diagonal sequence, and it is the negation of the diagonal sequence that is not in the list of sequences. Cantor's diagonal argument says nothing about the diagonal sequence itself, and there is no reason why it can't be in the list of sequences. Whether the diagonal sequence is or isn't in the list of sequences is not important to the diagonal argument.

Cantor states:

"Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v.

Thus, if av,v = m, then bv = w.

Then consider the element

E0 = (b1, b2, b3, …)"

My labeling the diagonal b may have been confusing. Will return to using D.

His negation E of the diagonal D depends on the existence of D. As explained in the previous post, there are no diagonal sequences in the list. They are horizontal as defined by Cantor. But the list shows E1as equal to D. Thus E1is an element of the list, so its negation E0 must also be an element of the list. That is a property of the binary set {0, 1.}

Although I agree about infinity being unbounded, are you sure that is the only property of infinity?

14th century. Via Old French from Latin infinitus , literally “not bounded,” from finitus “finished, finite.” vs. early European movies ending with 'el fin' (the end).

Immeasurable within the context of math. 

Measurement requires boundaries. You haven't explained how you would measure the length of a straight rod with only one end.

Cantor's theorem says otherwise. Indeed, Cantor's theorem implies that there are at least ℵ0 different magnitudes of infinity.

ℵ0 is the cardinality of the natural numbers. Perhaps you could tell me what the value of zero is.

He used ℵ0 to represent a value for 'infinite' sets which have no value.

The symbol 'π' has a value beginning with 3.14159.

The symbol 'e' has a value beginning with 2.7182.

The symbol '0' represents a set with no elements or {}.

 

 

KJW;

Congratulations on proving that N is infinite! But even though N is "inexhaustible", there still isn't enough elements in it to cover the set of all subsets of N.

I did not prove anything. The Peano axioms allow a perpetual method to form a greater integer n' = n + 1.

Any list of things can be counted using the set N. If not, what is the magic n where it fails?

Cantor makes this statement just before describing his diagonal method.

"However, there is a proof of this proposition that is much simpler, and which does not depend on considering the irrational numbers."

I stated from the beginning, my argument does not concern an application to numbers, only the sequence of symbols.

[KJW]

9 hours ago, phyti said:

My labeling the diagonal b may have been confusing. Will return to using D.

Thank you. Yes, your labelling the diagonal b was confusing.

 

 

9 hours ago, phyti said:

Cantor states:

Quote

"Then there is a series b1, b2, … bv,…, defined so that bv is also equal to m or w but is different from av,v.

Thus, if av,v = m, then bv = w.

Then consider the element

E0 = (b1, b2, b3, …)"

His negation E of the diagonal D depends on the existence of D. As explained in the previous post, there are no diagonal sequences in the list. They are horizontal as defined by Cantor. But the list shows E1 as equal to D. Thus E1 is an element of the list, so its negation E0 must also be an element of the list. That is a property of the binary set {0, 1.}

That E₁ is equal to the diagonal is coincidental and not part of the diagonal argument. Note that the given list of sequences is E₁, E₂, E₃,... etc. E₀ is not part of the list. E₀ is a sequence derived from the given list by negating the diagonal elements a_vv for v = 1, 2, 3,... etc. The diagonal argument is that E₀ cannot be an element of the given list, and therefore the given list must be incomplete with respect to all possible sequences.

 

 

9 hours ago, phyti said:

You haven't explained how you would measure the length of a straight rod with only one end.

I don't need to explain this because it is not relevant to the mathematics. And it is a glaring error on your part to think that it is.

 

 

9 hours ago, phyti said:

Quote

Perhaps you could tell me what the value of zero is.

The symbol 'π' has a value beginning with 3.14159.

The symbol 'e' has a value beginning with 2.7182.

Perhaps you can tell me what the values of 3.14159 and 2.7182 are.

 

 

9 hours ago, phyti said:

Quote

ℵ₀ is the cardinality of the natural numbers. Perhaps you could tell me what the value of zero is.

He used ℵ₀ to represent a value for 'infinite' sets which have no value.

The symbol '0' represents a set with no elements or {}.

I find it inconsistent that you use the cardinality of the set {} to specify the value of 0 while at the same time denying my use of the cardinality of the set [math]\textstyle \mathbb {N}[/math] to specify the value of [math]\aleph_0[/math].

 

 

9 hours ago, phyti said:

Any list of things can be counted using the set N. If not, what is the magic n where it fails?

Actually, you are correct... any list of things can be counted using the set [math]\textstyle \mathbb {N}[/math]. But it also reveals that you misunderstand the point of the diagonal argument. The point of the diagonal argument is that the set of all the possible sequences cannot be listed. That is, for every possible list of sequences, there will always be sequences that are missing from the list. And for any particular list, the diagonal argument identifies one of the missing sequences. Thus, we have at least two distinct infinities: an infinity that can be listed (countable), and an infinity that cannot be listed (uncountable).

 

 

9 hours ago, phyti said:

Cantor makes this statement just before describing his diagonal method.

"However, there is a proof of this proposition that is much simpler, and which does not depend on considering the irrational numbers."

I believe that Cantor was referring to an earlier proof that the cardinality of the real numbers is greater than the cardinality of the natural numbers which did not use the diagonal argument.

 

 

9 hours ago, phyti said:

I stated from the beginning, my argument does not concern an application to numbers, only the sequence of symbols.

Cantor's theorem is about a set, its elements, and its subsets.

 

 

Edited December 22, 2023 by KJW

[phyti]

 

 

[pzkpfw]

How many elements are in A?

How many elements are in B?

Is there a way to uniquely pair elements from A and B 1-to-1 ?

You are misapplying statistics.

[KJW]

4 hours ago, phyti said:

Earlier in this thread, you indicated that there is only one infinity. Have you changed your mind on that?

Anyway, in the case of y=x² and y=2x, they are the same infinity [math](\aleph_0)[/math]. The thing to note is that for every natural number x, there is one square x², and one double 2x, and that all the squares and doubles of all the natural numbers are different. That is, the mapping from all the natural numbers to their corresponding squares and doubles is exactly one-to-one. To say that there are not as many squares or evens as natural numbers is to say that there are natural numbers that do not have distinct squares or doubles, which is clearly false. Because a characteristic of infinite sets is that their elements can be mapped one-to-one onto proper subsets, the existence of a set of natural numbers that are not square or even is immaterial.

 

 

Edited December 22, 2023 by KJW

[studiot]

4 hours ago, pzkpfw said:

Is there a way to uniquely pair elements from A and B 1-to-1 ?

I'm sure you actually meant something else.