## Cantor Diagonal Argument disproof

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### Re: Cantor Diagonal Argument disproof

Jeff still keeps punishing himself by not coming to terms with the fact that the proposition we "disprove" by proof of negation is A 'the set S is countable' where it is already assumed that 'S is all of T', and which then of course leads to a contradiction (by using the lemma), or in Cantor's own words "that the anti-diagonal is both a part of T and not part of T", paraphrased; hence we conclude ~A.

Note that Jeff has never once disproved the above reasoning and evidence!

Note that Jeff has never once disproved the above reasoning and evidence!

- Gofer
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### Re: Cantor Diagonal Argument disproof

The difficulty is that both Gofer and JeffJo pick and choose what they want to respond to in a piecemeal way, and thereby ignore the overall argument.

If a string of finite significant characters is padded with (0), can a string of infinite significant characters be padded with (0)? E.g. Is it possible to have the string (1)(0)? I say this is impossible and we get (1). So if the generation of Tf goes to the limit, i.e. strings of infinite significant characters, there is no ...(0). Therefore the generation of Tf produces T in the limit.

If a string of finite significant characters is padded with (0), can a string of infinite significant characters be padded with (0)? E.g. Is it possible to have the string (1)(0)? I say this is impossible and we get (1). So if the generation of Tf goes to the limit, i.e. strings of infinite significant characters, there is no ...(0). Therefore the generation of Tf produces T in the limit.

- robert 46
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### Re: Cantor Diagonal Argument disproof

Robert, you are raising a straw man!

The overall issue is whether T is countable [1] or not. The actual theorem is ~A, where A is T is countable, which of course is proven using proof of negation, meaning we get to assume, for free, using the definition, that there exists a map (g n m) containing all of T, from which we then construct a new map, the "anti-diagonal", that can't be in m; hence we conclude ~A.

Cantor's proof is really just a visualization of the concept of maps, and can be seen as a proof sketch or an argument.

[1] according to the definition I just gave in my last post addressed to you

The overall issue is whether T is countable [1] or not. The actual theorem is ~A, where A is T is countable, which of course is proven using proof of negation, meaning we get to assume, for free, using the definition, that there exists a map (g n m) containing all of T, from which we then construct a new map, the "anti-diagonal", that can't be in m; hence we conclude ~A.

Cantor's proof is really just a visualization of the concept of maps, and can be seen as a proof sketch or an argument.

[1] according to the definition I just gave in my last post addressed to you

- Gofer
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### Re: Cantor Diagonal Argument disproof

Gofer still keeps punishing himself by continually changing what his claim is, in order to find the mistake he desperately wants me to have made, but can't seem to demonstrate.Gofer wrote:Jeff still keeps punishing himself by not coming to terms with the fact that the proposition we "disprove" by proof of negation is A 'the set S is countable' ...

The proposition we are ultimately trying to "disprove" is A&B, "the set S is countable and S is all of T." So right off the top, Gofer is lying in order to create a foundation for this new claim.

As Gofer has claimed in the past, we do that by tasking ourselves to prove B->¬A.

As Gofer has claimed in the past, we accomplish that by proving the lemma A->¬B, which is what Gofer has claimed in the past is accomplished via proof of negation[1].

And the contradiction he points to - if what Cantor meant was proof by contradiction[2] - follows from assuming S is all of T, not from assuming S is countable.

The fact that is is a proof by contraposition follows immediately from the point where Cantor says "From this proposition it follows immediately..."

Note that Jeff has never once disproved the above reasoning and evidence!

Note that I have repeatedly disproven the changing claims Gofer has made, which is why he keeps changing them. I didn't disprove this new one until now because, obviously, it is a new one.

+++++

[1] That by proving the lemma, and suddenly adding "Hey! What if we also assume B?", we can claim that this new assumption is the ϕ in Andrej Bauer's "To prove ¬ϕ, assume ϕ and derive absurdity." Then, since what the lemma proved is ¬B, the combination of assuming B and already having proven ¬B is a de facto absurdity. And the flaw in that claim, as I have repeatedly pointed out, is that in Andrej Bauer's "proof of negation," the absurdity has to be derived the assumption that B is true, not from that fact that we just proved that B can't be true when A is.

[2] Which Gofer still has not demonstrated must be called the different form, "proof of negation," nor admitted that made a trivial error when he named it improperly when he first made that claim.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

And robert has the gall to say I use projection?robert 46 wrote:The difficulty is that both Gofer and JeffJo pick and choose what they want to respond to in a piecemeal way, and thereby ignore the overall argument.

A simple perusal will show that I responded to all the points robert asked me to respond to. Then he picked, in a piecemeal way, the ones he felt he could refute. And I pointed out what was wrong in that.

What robert ignores here, is that he never created a string of infinite significant characters. He keeps glossing over the flawed step where he insists it must happen, ignoring all the reasons why it can't.If a string of finite significant characters is padded with (0), can a string of infinite significant characters be padded with (0)?

I say that question is irrelevant, until robert can come up with the algorithm he thinks produces this. I think he refuses to, because he knows I will point out the flaw in what he thinks happens.E.g. Is it possible to have the string (1)(0)?

Hint: The set N, of all natural numbers, is defined by induction. It contains the number 1 (the initialization step). And, if it contains the number n, it also contains the number n+1 (the induction step).

Every possible natural number is defined by this process; and by that, I mean there is nothing that can be called a natural number that is not. If you accept the Principle of the Excluded Middle[1], this defines entire the set. If you don't accept that the set is so defined, you have no business arguing about Cantor's theorem.

At any intermediate point in the process, after n entries, the count of the elements added to N is n. But the count of the members in the defined set is infinite. The abstract concept robert can't force himself to address, is that even though every intermediate count is an element of the set, the count of the entire set is not. There is no point in the induction process where a new member suddenly "morphs into" (robert's words) the value "infinity." Even the acceptance of the entire set does not make any member of it "morph into" the value "infinity." In fact, the property "infinity" can never be associated with, or compared to, any member of the set. Just the size of the set.

Similarly, there is no point in robert's claimed induction process (why can't he define it comprehensibly? It isn't hard) where he has a string of infinite significant characters. He only ever has a finite list of 1's and 0's, padded with infinite 0's.

+++++

[1] Which Gofer still hasn't demonstrated he understands.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:What robert ignores here, is that he never created a string of infinite significant characters. He keeps glossing over the flawed step where he insists it must happen, ignoring all the reasons why it can't.robert 46 wrote:If a string of finite significant characters is padded with (0), can a string of infinite significant characters be padded with (0)?

Then Cantor makes the same mistake. He considers the superset of a finite set of n characters, then leaps to the limit to consider the superset of a set of infinite characters. Furthermore, he leaps over the negation of the characters up to position n in row n as n increases to negating an infinite number of characters in infinite rows at the limit. If he can to it then why can't I?

- robert 46
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### Re: Cantor Diagonal Argument disproof

Here is the result of robert's last "in-a-piecemeal way" reply. As usual, I responded to all of what he said, refuting his overall argument. He ignored my overall argument.robert 46 wrote:The difficulty is that both Gofer and JeffJo pick and choose what they want to respond to in a piecemeal way, and thereby ignore the overall argument.

(robert's lies, and the corresponding facts, are color-coded.)

robert 46 wrote:JeffJo wrote:What robert ignores here, is that he never created a string of infinite significant characters. He keeps glossing over the flawed step where he insists it must happen, ignoring all the reasons why it can't.robert 46 wrote:If a string of finite significant characters is padded with (0), can a string of infinite significant characters be padded with (0)?

Then Cantor makes the same mistake. He considers the superset of a finite set of n characters, then leaps to the limit to consider the superset of a set of infinite characters.

Maybe if robert would read what Cantor wrote, he would recognize what Cantor did say, and didn't say. Then robert could stop making up lies about it.

Cantor never considered a finite set/string of n characters in his proof. Cantor never considered a finite set of strings in his proof, either. These are aspects that robert makes up, so that robert can claim he is extending Cantor's proof when robert considers a string of n characters, or a finite set of strings.

Cantor stated that one can specify a binary string of infinite length by indexing it to the known-to-be infinite set of natural numbers N, as in {x1, x2, ..., xn, ...}. He gave three examples, the equivalent of which, using robert's character's 0 and 1 since robert can't deign to use what Cantor did, were E1={0,0,0,0,...}, E2={1,1,1,1,...}, and E3={0,1,0,1,...}. Note that robert's method only finds one of these examples. This flaw is one of the cornerstones of the "overall argument" against robert's "extended analysis." One that robert ignores in his piecemeal replies.

You can't do what you say you want to do here, because you can't construct the "value" you call "infinity" by induction. You can construct a set that is infinite, and then use that set to define an infinite string. But that is not what you keep implying you are doing.Furthermore, he leaps over the negation of the characters up to position n in row n as n increases to negating an infinite number of characters in infinite rows at the limit. If he can to it then why can't I?

It is not the existence of of the anti-diagonal that is proven by induction, it is the existence of a function that defines every character in it. Cantor does not "leap over" anything, he created a way to define every character.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

Breaking News - Cantor's Lemma may prove non-constructive after all!

The definition of countability only guarantees a bijective map from S to some subset of N, and not its inverse which is implicitly used in the construction of the anti-diagonal in the Lemma.

The definition of countability only guarantees a bijective map from S to some subset of N, and not its inverse which is implicitly used in the construction of the anti-diagonal in the Lemma.

- Gofer
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:robert 46 wrote:JeffJo wrote:robert 46 wrote: If a string of finite significant characters is padded with (0), can a string of infinite significant characters be padded with (0)?

What robert ignores here, is that he never created a string of infinite significant characters. He keeps glossing over the flawed step where he insists it must happen, ignoring all the reasons why it can't.

Then Cantor makes the same mistake. He considers the superset of a finite set of n characters, then leaps to the limit to consider the superset of a set of infinite characters.

Maybe if robert would read what Cantor wrote, he would recognize what Cantor did say, and didn't say. Then robert could stop making up lies about it.

I extrapolated, so what if Cantor went right to the miracle directly? It is still based on the concept of extending from the finite into the realm of the infinite.

E1={0,0,0,0,...}, E2={1,1,1,1,...}, and E3={0,1,0,1,...}. Note that robert's method only finds one of these examples. This flaw is one of the cornerstones of the "overall argument" against robert's "extended analysis."

It finds all examples if the method can go to the limit of infinite significant characters from n significant characters.

You can't do what you say you want to do here, because you can't construct the "value" you call "infinity" by induction.Furthermore, he leaps over the negation of the characters up to position n in row n as n increases to negating an infinite number of characters in infinite rows at the limit. If he can to it then why can't I?

If he doesn't negate an infinite number of characters in infinite rows then Cantor has not produced ~D. Which is what I have been saying all along.

You can construct a set that is infinite, and then use that set to define an infinite string.

Nonsense. Infinite sets are hypothesized. If they could be constructed then they could be produced by a sequential process, but this requires going to the limit. Thus the infinite list would naturally precede the infinite set.

It is not the existence of the anti-diagonal that is proven by induction, it is the existence of a function that defines every character in it. Cantor does not "leap over" anything, he created a way to define every character.

The definition does not actually produce ~D. It keeps producing anti-diagonals of finite significant characters which become longer.

Notwithstanding that some people may believe in Cantor's sainthood, that doesn't prove that his miracle is not just a fast-talking con-job.

- robert 46
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### Re: Cantor Diagonal Argument disproof

Cantor never considers the "finite" of anything, making your statement here a straw man.robert 46 wrote: I extrapolated, so what if Cantor went right to the miracle directly? It is still based on the concept of extending from the finite into the realm of the infinite.

No, it doesn't, because every row in your constructed list will contain infinite strings of finite number of 1's. No row will ever contain infinite strings of infinite 1's and 0's.It finds all examples if the method can go to the limit of infinite significant characters from n significant characters.

That is exactly what Cantor's method does, that is "negating an infinite number of characters" ALL AT THE SAME "time".If he doesn't negate an infinite number of characters in infinite rows then Cantor has not produced ~D. Which is what I have been saying all along.

Robert, you are forgetting that we are in Cantor's world where unicorns actually do exist. You cannot bring a finitist argument into the world of infinite sets. But I can always bring constructive analysis into ANY world.Nonsense. Infinite sets are hypothesized. If they could be constructed then they could be produced by a sequential process, but this requires going to the limit.

Wrong again! Cantor's method is NOT sequential, meaning before going to position n+1, it must have passed position n, but happens all simultaneously.The definition does not actually produce ~D. It keeps producing anti-diagonals of finite significant characters which become longer.

- Gofer
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### Re: Cantor Diagonal Argument disproof

Nope, that's NOT what we're trying to do, because Cantor's Lemma already disproves that.JeffJo wrote: The proposition we are ultimately trying to "disprove" is A&B, "the set S is countable and S is all of T."

What we ARE trying to do is to show that if we assume B true, what does that imply for A? Well, it turns out that also assuming A leads to absurdity/contradiction with the help of the Lemma. Thus, we have proven that (A->~B)->(B->(A->false)), using proof of negation, namely the negation of A.

Jeff doesn't seem to comprehend that when proving something like B->~A, we get to assume B being true for free. Another way to look at it is that (A->~B) and B are both part of theory, and ONLY THEN do we assume A, which of course leads to absurdity/contradiction; hence, we conclude ~A. That is the proof of negation I'm talking about, regardless of Jeff lying or trying to misrepresent it.

- Gofer
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### Re: Cantor Diagonal Argument disproof

Gofer wrote:Cantor never considers the "finite" of anything, making your statement here a straw man.robert 46 wrote: I extrapolated, so what if Cantor went right to the miracle directly? It is still based on the concept of extending from the finite into the realm of the infinite.

Indeed he didn't, but this is just like the snake-oil salesman who claims his patent medicine cures what-ails-you, but glosses over what the ingredients are and how they work.

No, it doesn't, because every row in your constructed list will contain infinite strings of finite number of 1's. No row will ever contain infinite strings of infinite 1's and 0's.It finds all examples if the method can go to the limit of infinite significant characters from n significant characters.

I agree entirely that my method for producing strings of finite significant characters cannot get to the limit of producing strings of infinite significant characters because getting to the limit is an impossibility- there is no limit.

That is exactly what Cantor's method does, that is "negating an infinite number of characters" ALL AT THE SAME "time".If he doesn't negate an infinite number of characters in infinite rows then Cantor has not produced ~D. Which is what I have been saying all along.

Which is impossible because there is no explanation as to how Cantor's method of negating characters can get to the infinite limit either sequentially or all-at-once.

Robert, you are forgetting that we are in Cantor's world where unicorns actually do exist.Nonsense. Infinite sets are hypothesized. If they could be constructed then they could be produced by a sequential process, but this requires going to the limit.

Well then in my fantastic world, creating strings of finite significant characters presto-changeo becomes creating strings of infinite significant characters at the limit.

You cannot bring a finitist argument into the world of infinite sets.

Don't need to: the world of infinite sets is an obvious delusional fantasy which makes no more sense than Carroll's Alice in Wonderland- hedgehogs for croquet balls and flamingos for mallets. Just try to play a game of croquet under those conditions.

But I can always bring constructive analysis into ANY world.

Not where the methods do not work under the circumstances.

Wrong again! Cantor's method is NOT sequential, meaning before going to position n+1, it must have passed position n, but happens all simultaneously.The definition does not actually produce ~D. It keeps producing anti-diagonals of finite significant characters which become longer.

All Cantor does is claim to have a method, but conveniently fails to show that the method actually works: it cannot merely be defined to work- that is a con-job for the easily duped. Perhaps Cantor was the greatest snake-oil salesman ever to come to mathematics.

- robert 46
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### Re: Cantor Diagonal Argument disproof

Gofer wrote:Nope, that's NOT what we're trying to do, ...JeffJo wrote: The proposition we are ultimately trying to "disprove" is A&B, "the set S is countable and S is all of T."

Cantor states he is seeking "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."

And we know what that is, immediately, by contraposition.What we ARE trying to do is to show that if we assume B true, what does that imply for A?

Gofer doesn't seem to comprehend that this doesn't need saying, and that I understand it fully.Jeff doesn't seem to comprehend that when proving something like B->~A, we get to assume B being true for free.

What Gopher ignores, is that this is the identifying property that [url=wikipedia.org/wiki/Contraposition]differentiates Proof by Contraposition from Proof by Contradiction[/url]

What Gofer can't even bring himself to address, is that all of the forms he claims require this "free assumption" to be a precursor to the contradiction/absurdity that results. Not the actual expression of it.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

Right, which obviously is the main theorem, and is NOT what you stated ~(A and B). But do you notice something? The main theorem has exactly the form ~P, for which proof of negation is particularly well suited, making Cantor's Lemma unnecessary.JeffJo wrote:Cantor states he is seeking "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."

and since contraposition is a THEOREM proven using proof of negation ...And we know what that is, immediately, by contraposition.What we ARE trying to do is to show that if we assume B true, what does that imply for A?

Jeff seems to be hung up on the fact that the deduction doesn't fit the form of ~P because of the logical connective ->; this is quite wrong however.

- Gofer
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### Re: Cantor Diagonal Argument disproof

The "ingredients" in this case is the set of infinite strings; and the "snake oil" is the belief that this set is countable; the "cure" is Cantor's proof.robert 46 wrote:Indeed he didn't, but this is just like the snake-oil salesman who claims his patent medicine cures what-ails-you, but glosses over what the ingredients are and how they work.

Yes there is, like we have been telling and yelling at you: FUNCTIONS. The function (2*) doesn't go to any limit, yet is still valid "simultaneously" for all numbers.I agree entirely that my method for producing strings of finite significant characters cannot get to the limit of producing strings of infinite significant characters because getting to the limit is an impossibility- there is no limit. [...]

Which is impossible because there is no explanation as to how Cantor's method of negating characters can get to the infinite limit either sequentially or all-at-once.

You could do that; but it still has nothing to do with what Cantor does.Well then in my fantastic world, creating strings of finite significant characters presto-changeo becomes creating strings of infinite significant characters at the limit.

So then you believe that the function (2*) only has a finite domain? And dare we never speak of the "numbers as a whole" for which it is defined?Don't need to: the world of infinite sets is an obvious delusional fantasy which makes no more sense than Carroll's Alice in Wonderland- hedgehogs for croquet balls and flamingos for mallets. Just try to play a game of croquet under those conditions.

What do you mean with "show that the method actually works"? Cantor's shows that the anti-diagonal of any enumeration can't be part of the enumeration; if it were, we'd have the contradiction Cantor mentions, "that a thing would be both part and not part of T.", paraphrased.All Cantor does is claim to have a method, but conveniently fails to show that the method actually works: it cannot merely be defined to work- that is a con-job for the easily duped. Perhaps Cantor was the greatest snake-oil salesman ever to come to mathematics.

- Gofer
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