# problem with cantor diagonal argument

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Ann and Bill independently work on examples of Cantor's diagonal argument.

Ann:

111001...

000111...

101100...

110011...

000110...

010110...

transforms diagonal D1 to alternating '01' sequence T1,

010101...
which can't appear in any list per the cda.

Bill:

000110...

010110...

110001...

000111...

101100...

110011...

transforms diagonal D2 to alternating '10' sequence T2,

101010...

which can't appear in any list per the cda.

If D1 and D2 appear in any list, they must be members of the complete list.
T1=D2.

I.e. a missing sequence is only relative to an individual list.

Edited by phyti

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5 minutes ago, phyti said:

which can't appear in any list per the cda

There is no problem, because the cda does not say that a constructed sequence does not appear in any list, but in the given list.

Quote

If s1, s2, ... , sn, ... is any enumeration of elements from T, then an element s of T can be constructed that doesn't correspond to any sn in the enumeration.

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That's what I said.
"I.e. a missing sequence is only relative to an individual list".

Here is what Cantor said.

"If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E of M, which cannot be connected with any element Ev."

"From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M."

From the example T1 which is missing from Ann's list is included as D2 in Bill' list.
Thus it is not missing from M.
Missing from a subset does not imply missing from the complete set.
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On 5/24/2023 at 12:24 PM, phyti said:

That's what I said.
"I.e. a missing sequence is only relative to an individual list".

Here is what Cantor said.

"If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E of M, which cannot be connected with any element Ev."

"From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M."

From the example T1 which is missing from Ann's list is included as D2 in Bill' list.
Thus it is not missing from M.
Missing from a subset does not imply missing from the complete set.

CDA is about cardinality (size) of the two sets (countable vs uncountable).  I am not sure exactly what the problem you have with CDA.

I am thinking about an analogy to your issue to see what happens if we apply your issue on a much smaller scale.

Assume Bill and Ann want to do the same thing with a set L vs a smaller set S.  They want to know which set is bigger, but they don't want to count the elements.   I will keep this in binary-type numbers vs natural numbers

Let's say L = {000, 111, 110, 011, 100, 001, 010}, and S = {2, 4, 5}.

(Keep in mind that the correlated elements don't have to be same things. K = {5, 7, 1} has three elements and P = {@, U, 9} also has three elements; their cardinalities are the same).

Ann might get T1 = 101 and Bill might get T2 = 110.  The set L can still be larger than the set S even though T1 doesn't equal T2.

And clearly, we can always find an element of L that can not match to an element of S.

I see no issue here.  Or am I missing your point?

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;

"If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0 of M, which cannot be connected with any element Ev."

"From this proposition it follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M."

Cantor believes there is a hierarchy of infinities, a communication from above which he is obligated to explain to the world.
Cantor is selling his idea of transfinite numbers to anyone who will listen.
In the above quotes he states there are always elements of M which differ from those in any list, thus there cannot be a one to one correspondence with the set of natural numbers N, even though N is inexhaustible.

The Ann and Bob example contradicts his idea, since one will have an infinite list containing the sequence E0 that the other infinite list does not contain.

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43 minutes ago, phyti said:

In the above quotes he states there are always elements of M which differ from those in any list, thus there cannot be a one to one correspondence with the set of natural numbers N, even though N is inexhaustible.

The Ann and Bob example contradicts his idea, since one will have an infinite list containing the sequence E0 that the other infinite list does not contain.

No, there is no contradiction. As per Cantor, for any list there is an element in M which differs from those in that list.

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IOW, Cantor does not say,

∃ element∈M, ∀ list L, element∉L

He says,

∀ list L, ∃ element∈M, element∉L

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"If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0 of M, which cannot be connected with any element Ev."

E0 is the negation of D, the diagonal used to form E0. The negation of a sequence has all positions with m and w interchanged. That is the connection.

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Using a finite set of 10 unique sequences, there are

10 possible S for position 1,

9 possible S for position 2,

8 possible S for position 3,

...

1 possible S for position 10.

A total of 10! random lists, to show

there is not just 1 random list L for the sequences in Cantor's set M.

Each L begins with a different sequence, thus a different diagonal D.

One of those L's will contain a diagonal E0 that Cantor declared missing from the L containing the original D.

Cantor makes his own contradiction by ignoring the complementary nature of a sequence and its negation, i.e. they come in pairs for a binary set of 2 symbols.

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• 2 weeks later...
On 5/23/2023 at 12:27 PM, phyti said:

Ann and Bill independently work on examples of Cantor's diagonal argument.

On 5/23/2023 at 12:27 PM, phyti said:

If D1 and D2 appear in any list, they must be members of the complete list.

T1=D2.

I.e. a missing sequence is only relative to an individual list.

Why do you think D1 and D2 appear in any list that is used as an "example of CDA"? Are you still under the impression that CDA starts with a statement like this?

• Assume you are "working with" a list of the complete set of infinite binary sequences.

That impression is understandable, for one who has not actually read the paper where Cantor presented CDA. It is commonly claimed to be the first line.

But you have read it; I know this, because I saw you quote it in a different post. The proposition he proves is (in literal form and an informal translation in the style you use for Ann and Bill) :

• If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0­ of M, which cannot be connected with any element Ev.
• If Ann "works with" an infinite list of infinite binary sequences, then there is always an infinite binary sequence a0 that is not in Ann's list.

To prove this, Cantor starts out by saying:

• 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, …), ... , where the characters au,v are either m or w.

• Sorry for the copy-and-paste text, I got it straight from Logicmuseum.

• For proof, let Ann start out with a list like 111001..., 000111..., 101100..., ...

Now please, please notice that neither of the statements says "assume that every infinite binary sequence is listed." They just have to be lists that actually do exist. Cantor did make one mistake here - technically, he should prove (by example is sufficient) that at least one such list is possible. You didn't imply how Ann and Bill determine any more members. But here's one:

• 100000...
• 010000...
• 001000...
• ...

So, let's go back to my question at the top. Why do you think D1 and D2 appear in any list? Yes, they should appear in any complete list, but what has that to do with Ann's and Bill's lists?

The sequence a0 (that you call D1) does not appear in Ann's list. Whether or not it appears in Bill's is irrelevant - they afr not necessarily listing the same set of sequences. The sequence b0 (that you call D2) does not appear in Bill's list. Whether or not it appears in Ann's is also irrelevant. But we can follow this with the conclusion of CDA, which is actually the contradiction that you tried to use to claim it was invalid:

• It follows immediately that the totality of all elements of M cannot be put into the sequence [Reihenform]: E1, E2, …, Ev, … otherwise we would have the contradiction, that a thing [Ding] E0 would be both an element of M, but also not an element of M.
• It follows immediately that Ann (or Bill) could not have been "working with" a listing of all infinite binary sequences. Because that would mean that the string a0=D1 (or b0=D2) was both in the set listed by Ann (or Bill), but also not in that set.

CDA actually isn't very hard to understand. It is unfortunate that it gets taught incorrectly, becasue that does get in the way of understanding it. But you can't disprove it by intentionally ignoring what it says, and using its contradiction against itself.

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Jeff;
After a few failed attempts, Cantor's error is exposed.
Keep in mind the nature/properties of a random list, a 1-dimensional sequence, and a 2-dimensional sequence.
Read the pdf without any preconceived notions.

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On 6/16/2023 at 12:50 PM, phyti said:

Jeff;
After a few failed attempts, Cantor's error is exposed.
Keep in mind the nature/properties of a random list, a 1-dimensional sequence, and a 2-dimensional sequence.

I have read it, without preconceptions. It is very confused. Maybe you should read Cantor's proof without your preconceptions.

From the pdf: "He [Cantor] concludes E0 differs from D for all v in the set M"

No, he concludes that E0 differs from Eu for all u in the list that E0 was derived from. Neither "v" nor "u" are in the set M, they are natural numbers not binary sequences (rows). What Hudson calls "D" is the diagonal that is negated to get Cantor's E0. Both are in M, by the definition of M. And nothing described as "all <whatever> in M" are referenced in this part of the proof.

The rest of the paper refers to constructions that have nothing to do with this conclusion, so they are irrelevant. Whether or not "A different random list could contain E0" is irrelevant, since any list you can produce can be used to construct a different E0 that is not in that list.

And the point is really quite simple: The actual diagonal argument does not prove that M is uncountable, it proves:

• IF you have a list of sequences from M, THEN we can construct a sequence that is in M, but not in this list.

In "analyzing" this proposition, there is no reason to refer to "all M," or any other sequences besides 00 derived from the list in hand. But once you have the truth of this proposition, it "follows immediately" that M itself is uncountable.

Edited by JeffJo
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4 hours ago, JeffJo said:

I have read it, without preconceptions. It is very confused. Maybe you should read Cantor's proof without your preconceptions.

Hello Jeffjo.  and welcome

Good to see some good solid maths being promoted +1

I'm (half) sure I've seen you on some other maths forums ?

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On 6/17/2023 at 6:10 PM, studiot said:

I'm (half) sure I've seen you on some other maths forums ?

Yeah, I have several Quixotic causes where I try to un-convince people who are convinced of an incorrect solution. Another here is the explanation of rainbows that is almost universally taught, yet contains errors of fact and interpretation.

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Jeff;

Cantor wanted to prove an infinite list/enumeration of sequences contained in his set M was greater than the infinite set N of natural numbers, thus no 1 to 1 correspondence and requiring his transfinite numbers. Showing incomplete partial lists would not accomplish his goal.

His two quotes indicate his goal.

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

M is the set of all E sequences.

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

M is greater in number of elements than the set N of natural numbers.

You: "Neither "v" nor "u" are in the set M, they are natural numbers not binary sequences (rows)"
No one said that, so stop reading things into the post that aren't there.
Cantor uses a u,v coordinate system to locate any symbol in a 2 dimensional form of a list. Example 1.

An excerpt from Cantor's philosophical writing:
1. As for the mathematical infini

2. te, to the extent that it has

3. found a justified application

4. in science and contributed to

5. is usefulness, it seems to me

6. that it has hitherto appeared

7. principally in the role of a

8. variable quantity, which eith

9. er grows beyond all bounds or

10.diminishes to any desired minu

11.teness, but always remains fin

12.ite.  I call this the improper

The red diagonal D = Aeusu_ie_stl, where _ is a space.
The example contains 12 statements formed using rules of grammer.
Its purpose is to show D is not a statement just because it originates from a list of statements, only a random sequence of symbols, formed by a simple rule of selection. Each symbol in D has the coordinates (u, u).
The rules of formation or definition are different for the list and the diagonal.

In Cantor's argument the initial list contains randomly entered E sequences composed of two symbols. Each E sequence is 1-dimensional and extends without limit horizontally with a varying coordinate v, (1, 2, 3, ...)
The diagonal D has the coordinates (u,u), using the same rule of selection.
D is not an E just because it originates from a list of E's.
The rules of formation or definition are different for the E list and the diagonal D.
The D sequence is also 1-dimensional but extends horizontally and vertically, requiring two coordinates (u, v) for each position.
The difference between D and E requires an analysis of the small details which apparently were never done. Another case of evidence in plain sight, the most difficult to detect.

Accepting D and E as equivalent, a problem appears where they intersect as in the pdf fig.3. An E and a D can occupy the same position, only if the symbol is the same for both. For the negation of D, E0 obviously cannot appear anywhere in the list. Not because of any oversight, but exclusion resulting from D, and that only because D is the first in the list.
There are other exclusions. If E has 1 symbol in column v that differs from D, it is excluded from that row. If E has 2 symbols in columns v1 and v2 that differ from D, it is excluded from those 2 rows. In general if E has k columns with symbols differing from D, E is excluded from those rows.

His diagonal imposes an order on a random list, which has no order.
I include the above to show how he convinced himself and others while neglecting the properties of a random list. Using the term 'contrived diagonal' is not implying he intended to deceive anyone.

What would prevent a random sequence from being inserted anywhere in a random list?
As shown in the pdf, if the diagonal began at row 2, then E0 could appear at row 1.

You are confused because you didn't see what you wanted to see.
Read his biography and philosophical thoughts, and his correspondence with his contemporaries. Many disagreed with him. One of my favorites:
Leopold Kronecker: "I don't know what predominates in Cantor's theory - philosophy or theology, but I am sure that there is no mathematics there."

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5 hours ago, phyti said:

Jeff;

Cantor wanted to prove an infinite list/enumeration of sequences contained in his set M was greater than the infinite set N of natural numbers, thus no 1 to 1 correspondence.

Maybe you should come back when you learn to read. He didn't need to imply what he wanted to prove, he stated it explicitly.

"[In 1875] there appeared, probably for the first time, a proof of the proposition that there is an infinite manifold, which cannot be put into a one-one correlation with the totality [Gesamtheit] of all finite whole numbers 1, 2, 3, …, v, …, ... However, there is a proof of this proposition that is much simpler."

And you should at least attempt to understand what the words you ignore mean. A infinite list is, by definition, a set that is in a bijection with the set N of natural numbers. That means it has the same cardinality (oops, you left that one out), not a greater cardinality. He wanted to prove that the cardinality of M was greater, not "an infinite list/enumeration of sequences contained in M."

5 hours ago, phyti said:

You: "Neither "v" nor "u" are in the set M, they are natural numbers not binary sequences (rows)"

No one said that, so stop reading things into the post that aren't there.

Maybe you should read the .pdf you linked, without your pre-existing bias. I call your attention to the section titled "Cantor's method", in the paragraph following figure 1, line 4: "for all v in the set M." But "v" is the index of the columns, while he meant the index "u" of the rows.

5 hours ago, phyti said:

The diagonal D has the coordinates (u,u), using the same rule of selection.
D is not an E just because it originates from a list of E's.

D has the single coordinate u. It is a function whose domain is the set of natural numbers, and whose co-domain is {'m','w'}.

The set M is the set of all functions whose whose domains are the set of natural numbers, and whose co-domains are {'m','w'}.

So by definition, D is in M. That is, it is an "E".

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Jeff;

His proof using the diagonal is flawed for the reasons mentioned.

If a sequence D is a member of M or any binary set of symbols,

then its negation must also be a member of M.

You are in denial.

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On 5/23/2023 at 10:27 AM, phyti said:

I.e. a missing sequence is only relative to an individual list.

I think we can all agree on this.  I don't understand what the problem is.

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Boltz;

Anyone can copy Cantor’s method and get the same result of the ‘missing’ sequence which is misdirection similar to what the illusionist does.

He defined two different types of sequences, horizontal and diagonal, mixed them in the same list and fooled himself into thinking he proved his need for transfinite numbers.

S1 010101

S2 _1

S3__1

S4___1

S5 ____1

_ is either 0 or 1
In the above list, which sequence, horiz or diag, gets the id of S1?

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5 hours ago, phyti said:

Boltz;

Anyone can copy Cantor’s method and get the same result of the ‘missing’ sequence which is misdirection similar to what the illusionist does.

He defined two different types of sequences, horizontal and diagonal, mixed them in the same list and fooled himself into thinking he proved his need for transfinite numbers.

S1 010101

S2 _1

S3__1

S4___1

S5 ____1

_ is either 0 or 1
In the above list, which sequence, horiz or diag, gets the id of S1?

There is nothing wrong with Cantor's argument.

It is called diagonalisation because it is usually presented in grid format.

But this format is not necessary, The issue arises because the set of all sets of positive integers in not enumerable.

That is it cannot be put into one-to-one correspondence with the set of all positive integers.

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On 6/21/2023 at 11:31 AM, phyti said:

If a sequence D is a member of M or any binary set of symbols,

then its negation must also be a member of M.

You are in denial.

Why? It is you who are in denial. First off, I'm not quite sure you understand what the sets are. As evidence, "any set of binary symbols" seems to be a vacuous description.

Cantor's set M (as opposed to whatever you think M is) is the set of all possible Cantor Strings (my name). A Cantor String is a function C that maps the set N of all natural numbers, starting with 1, to the set {0,1}. (Well, Cantor used {'m','w'}, but any difference is insignificant.) We can write this C:N->{0,1}. Any individual character in this string can be expressed as C(n), for any n in N.

Cantor's Diagonal Argument does not use M as its basis. It uses any subset S of M that can be expressed as the range of a function S:N->M. So any individual string in this function can be expressed as S(n), for any n in N. And the mth character in the nth string is S(n)(m).

So the diagonal is D:N->{0.1} is the string where D(n)=S(n)(n). By definition, this is a Cantor String, so it is in M. And the negation of it, D', is the string where D'(n)=1-S(n)(n). And it also, by definition, is in M.

What isn't necessarily true, is that either D or D' is in S. D may or may not be, but that is irrelevant. D' cannot be, because D' is different than every S(n) in the nth position.

On 6/23/2023 at 12:44 PM, phyti said:

[Cantor] defined two different types of sequences, horizontal and diagonal...

No, he did not. You did. Cantor defined functions of the form C:N->{0,1} and showed that D' fit that form. How the characters are extracted from the 2D table used to illustrate it has no significance.

Edited by JeffJo
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studiot;

Quote

The issue arises because the set  of all sets of positive integers in not enumerable.

Isn't that is about power sets?

This is about the method of using the diagonal in a random list, not about numbers in any form. He used alphabetical symbols in his cda, not numbers!

Jeff;

1. As evidence, "any set of binary symbols" seems to be a vacuous description.

2. A Cantor String is a function C that maps the set N of all natural numbers, starting with 1, to the set {0,1}. (Well, Cantor used {'m','w'}, but any difference is insignificant).

It's possible you have a problem in comprehension.

Read the beginning of his argument.

1. Cantor defines his set M as the totality of all elements E,

where each E contains an infinite (unlimited) number of characters (symbols), selected from a binary set of two, {m, w}.

He "maintains" (declares) the set M "does not have the power" (cardinality) of N the set of integers.

2. Cantor is not using subsets or functions. He defined all the elements needed.

Cantor is comparing two sets M and N, by attempting a 1 to 1 correspondence. If that fails, then he can sell his transfinite numbers.

He begins with the standard 2-dimensional array for three E sequences/strings.

He defines a diagonal D, and in the process forms a duplicate sequence and alters the random array with an ordering rule. In a random list, there is no order. Each row can appear anywhere within the list, thus its name!

Why didn't his negation E0 inherit the same 45º angle from diagonal D? Then they could coexist as shown in fig.2 of the pdf.

Here is a phrase from his philosophy on set theory

"if alpha is an infinitely large number,"

What is the meaning of the red if any?

Another of my favorite quotes.

Wittgenstein: "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".

Edited by phyti
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23 hours ago, phyti said:

studiot;

Isn't that is about power sets?

I suspect there are German words that mean the following, but I only know a little German ("he's sitting over there" - the movie Top Secret). But this is why I use '0' and '1' as the characters.

Take '1' to mean "true" or "included." Take "0" to mean "false" or "excluded." Then each Cantor String - a function of the form C:N->{'0','1'} - defines a unique subset of the natural numbers by including or excluding its position number. CDA is not a formal proof, it is an example, using infinite sets, of Cantor's Theorem that the power set of any set has a greater cardinality than that set.

Quote

It's possible you have a problem in comprehension.

It's obvious that you don't want to comprehend CDA. You want to misinterpret it in your failed attempt to disprove it. I fully comprehend the parts of your argument that are comprehendable. Most aren't.

Quote

1. Cantor defines his set M as the totality of all elements E,

And I use C:N-->{'0','1'} in place of E. This is a properly-define function that map every natural number n to one of two characters. The set M (I often use "T" go be consistent with Wikipedia's article, but I don't recall if I have done so here) is the set of all such functions.

Quote

2. Cantor is not using subsets or functions. He defined all the elements needed.

On the contrary, every E he uses is a function, as I showed. He didn't define the ones used in CDA, he defined the conditions they must satisfy.

Quote

Cantor is comparing two sets M and N, by attempting a 1 to 1 correspondence. If that fails, then he can sell his transfinite numbers.

Cantor never compared N to his set M. This is a common misconception, but it is a misconception. And if you have read the actual paper, this statement alone proves that you did not "comprehend" it. You don't even get all the words right.

Cantor says that there are subsets of M that can be put into a one:one correlation (he means a bijection, while "correspondence" means an injection). This is true, there are trivial examples like "all '0' except a '1' in position n." He goes on to prove that any example that can exist misses at least one member of M.

Quote

He begins with the standard 2-dimensional array for three E sequences/strings.

He does no such thing. Those three strings were examples of what the strings in M can look like, not the beginning of the CDA proof. He hasn't even presented the proposition that CDA proves at that point in the paper. Which is:

3. "If E1, E2, …, Ev, … is any simply infinite series of elements of the manifold M, then there always exists an element E0­ of M, which cannot be connected with any element Ev.

This "begins with" a set of E's - not the totality of all E's which comprise M - that has the property that it is in a bijection with N.

Quote

He defines a diagonal D ...

He does no such thing. Nothing like your "D" is mentioned in the paper.

Quote

... and in the process forms a duplicate sequence and alters the random array with an ordering rule. In a random list, there is no order. Each row can appear anywhere within the list, thus its name!

He does no such thing. There is nothing "random," tho it is "arbitrary." The proof "begins with" any ordered list E1, E2, ..., Ev, ... that can exist. Each Ev is defined in this list, but "arbitrary" means that the actual definition doesn't matter.

Quote

Why didn't his negation E0 inherit the same 45º angle from diagonal D? Then they could coexist as shown in fig.2 of the pdf.

Because Cantor never mentions "diagonal," never defines a "diagonal D," nor a "45º angle." He defines a new function where E0(n) = 1-E(n,n). Later Mathematicians interpret this as a diagonal, but that term is not mathematically defined in the paper, nor is it pertinent to any part of the proof.

This new function E0(n) = 1-E(n,n) fits my definition of a Cantor String C:N->{'0','1'}, so it is a member of M.

Quote

Here is a phrase from his philosophy on set theory

Ignoring the fact that this quote has nothing to do with CDA, so it is irrelevant to CDA, you fail to "comprehend" it. You just don't like what it means.

Edited by JeffJo
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23 hours ago, phyti said:

Read the beginning of his argument.

Can you link the version of Cantor's proof that you're looking at? I looked up this translation: https://www.jamesrmeyer.com/infinite/cantors-original-1891-proof.php and could not correspond it to your exposition.

23 hours ago, phyti said:

"if alpha is an infinitely large number,"

Just to pick one example, this phrase appears nowhere in the translation I linked.

Edited by wtf
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Jeff;

Keeping it simple, using 0 and 1 for symbols and S for sequence or string, divide M into 2 subsets,
M0 containing S beginning with '0', and M1 containing S beginning with '1'.
S=0000... is a member of M0, true or false?
S=1111... is a member of M1, true or false?

3 hours ago, wtf said:

Can you link the version of Cantor's proof that you're looking at? I looked up this translation: https://www.jamesrmeyer.com/infinite/cantors-original-1891-proof.php and could not correspond it to your exposition.

Just to pick one example, this phrase appears nowhere in the translation I linked.

"if alpha is an infinitely large number,"

From CANTOR'S PHILOSOPHICAL WRITING

CDA

Both from

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