## Cantor Diagonal Argument disproof

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### Re: Cantor Diagonal Argument disproof

Gofer wrote:No, that would be what Robert is really trying to say but can't seem to comprehend that that is all he's saying; and that is not what Cantor's theory says.Edward Marcus wrote:So is Cantors theory a fancy way of telling us we cannot count to infinity?

Cantor is saying there is something bigger than endless: that is nonsensical. Gofer doesn't understand what I am saying, so that discounts his diversions.

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

No, Cantor is saying that, according to the definition of cardinality, N is less than T.robert 46 wrote: Cantor is saying there is something bigger than endless: that is nonsensical.

More like, Robert doesn't comprehend what I am saying, hence his diversions and continued avoidance of my previous post regarding algorithms.Gofer doesn't understand what I am saying, so that discounts his diversions.

Robert's whole argument is that, after having examined n rows, there are 2^n-n strings unaccounted for. Robert of course misses the elephant in the room, namely that, after having examined n strings, there are an INFINITY of strings left, reducing his argument to exactly that.

Robert doesn't seem to comprehend that, from the algorithmic viewpoint, to successfully enumerate a set, it suffices if every one of its members will be calculated in finite time, by the algorithm, which, in the case of the SFSC's, there is such an algorithm. Of course, that algorithm will never finish, but that is not the point; so if Robert argues that it should, he's misrepresenting algorithmic enumerability.

- Gofer
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### Re: Cantor Diagonal Argument disproof

Gofer wrote:No, Cantor is saying that, according to the definition of cardinality, N is less than T.robert 46 wrote: Cantor is saying there is something bigger than endless: that is nonsensical.

So, Cantor is saying that T and N are both endless, but T is more endless than N? That also is nonsensical. "Cardinality" is a manufactured term which has no more meaning than what the academicians wish to ascribe to it.

More like, Robert doesn't comprehend what I am saying, hence his diversions and continued avoidance of my previous post regarding algorithms.Gofer doesn't understand what I am saying, so that discounts his diversions.

Another parrot. We have established that diagonalization has no algorithm which leads to a convergence.

Robert's whole argument is that, after having examined n rows, there are 2^n-n strings unaccounted for. Robert of course misses the elephant in the room, namely that, after having examined n strings, there are an INFINITY of strings left, reducing his argument to exactly that.

No, I am saying that after n columns are examined, there are 2^n-n more rows to examine, which in turn means there are 2^n-n more columns to examine, which means.... We can see that this is an endless process where one gets further behind in the examination compared to what is required. The conclusion to reach is that there is no conclusion to an endless process, and that means there is no ~D which is defined.

Robert doesn't seem to comprehend that, from the algorithmic viewpoint, to successfully enumerate a set, it suffices if every one of its members will be calculated in finite time, by the algorithm, which, in the case of the SFSC's, there is such an algorithm. Of course, that algorithm will never finish, but that is not the point;

That is exactly the point because ~D would come after the conclusion of the algorithm.

so if Robert argues that it should, he's misrepresenting algorithmic enumerability.

I don't question the enumerability of the IADs, but that ~D is not one of them.

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

If I had a say,I would ban these infinity operations

inf = ,inf<, inf x , inf / , / inf, etc...

inf = ,inf<, inf x , inf / , / inf, etc...

- Edward Marcus
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### Re: Cantor Diagonal Argument disproof

No.robert 46 wrote:So, Cantor is saying that T and N are both endless, but T is more endless than N? That also is nonsensical. "Cardinality" is a manufactured term which has no more meaning than what the academicians wish to ascribe to it.

Cantor is saying that N cannot be put into a 1:1 correspondence with T, but can be put into a 1:1 correspondence with a subset of T. He calls this relationship "cardinality," and says that T has the greater cardinality.

The only nonsensical concept here is when people like you, who refuse to even try to understand what it does, and does not, mean try to make it about numerical value, size, or degree of "endlessness". It is an abstract concept, like "color" of particles in quantum mechanics, and is not supposed to have real-world meaning.

... and when you say that, you are ignoring the fact the anything you conclude can only apply to finite lists of finite strings. You are ignoring the fact that your arguments actually require an end to the list, and the strings. You ASSUME THERE IS AN END, and from that conclusion conclude there must be an end.No, I am saying that after n columns are examined, there are 2^n-n more rows to examine...

The IADs are, by definition, enumerable since they are defined by a finite portion of an ennumerable list. ~D is not one of them, because it is defined by an infinite list. All your procedure proves, is that if you require an end, there must be an end.I don't question the enumerability of the IADs, but that ~D is not one of them.

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:No.robert 46 wrote:So, Cantor is saying that T and N are both endless, but T is more endless than N? That also is nonsensical. "Cardinality" is a manufactured term which has no more meaning than what the academicians wish to ascribe to it.

Cantor is saying that N cannot be put into a 1:1 correspondence with T, but can be put into a 1:1 correspondence with a subset of T. He calls this relationship "cardinality," and says that T has the greater cardinality.

The only nonsensical concept here is when people like you, who refuse to even try to understand what it does, and does not, mean try to make it about numerical value, size, or degree of "endlessness". It is an abstract concept, like "color" of particles in quantum mechanics, and is not supposed to have real-world meaning.

It is nothing but fantasy.

... and when you say that, you are ignoring the fact the anything you conclude can only apply to finite lists of finite strings. You are ignoring the fact that your arguments actually require an end to the list, and the strings. You ASSUME THERE IS AN END, and from that conclusion conclude there must be an end.No, I am saying that after n columns are examined, there are 2^n-n more rows to examine...

I have proved that all the IADs must be produced and found before ~D could be produced. We have agreed that ~D is not consequent to convergence. To refer back to Gauss:

'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.

The IADs are, by definition, enumerable since they are defined by a finite portion of an ennumerable list. ~D is not one of them, because it is defined by an infinite list. All your procedure proves, is that if you require an end, there must be an end.I don't question the enumerability of the IADs, but that ~D is not one of them.

With Gauss and others for backup, it should be clear that Cantorian set theory is a silly fantasy which for those who want to believe is religious doctrine.

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

Only in your mind!I have proved that all the IADs must be produced and found before ~D could be produced.

Robert doesn't comprehend that the whole of ~D need not be produced in order to prove it can't equal any row; to prove it for row N, we only need to calculate the N'th bit of ~D; and since N is arbitrary, it must hold for all N's, thus rebutting Robert's statement.

Of course, if we instead use function theory, ~D is constructed all at once, through g(n,n)+1 mod 2, so that that function does not equal any function of any row.

- Gofer
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### Re: Cantor Diagonal Argument disproof

It is an abstract concept that follows from definitions and axioms, just like equilateral triangles and the determinant of a matrix. If you want to call these fantasies, then go right ahead. They still have mathematical meaning.robert 46 wrote:It is nothing but fantasy.

But calling it a fantasy is the way you evade points you can't argue against, so it really carries no weight.

You think you have proven that, if a sequential algorithm is required, that all the IADs must be found before the only thing we want, the anti-diagonal. You are wrong to think so, because you deny the Axiom of Infinity, and you haven't established why it needs to be algorithmic. So all you have really done, is blow hot air.I have proved that all the IADs must be produced and found before ~D could be produced.

No, you changed what we said into that so you can misinterpret. What we said, is that convergence applies to completely different things, and is irrelevant to finding the anti-diagonal.We have agreed that ~D is not consequent to convergence.

No. He was talking about a completely different kind of infinity, potential infinity. Anything having to do with it is irrelevant to Cantor. Which is also the failing of everybody else you quote.Refer back to Gauss:

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:It is an abstract concept that follows from definitions and axioms, just like equilateral triangles and the determinant of a matrix. If you want to call these fantasies, then go right ahead. They still have mathematical meaning.robert 46 wrote:It is nothing but fantasy.

That a proper subset can be as large (equinumerous) as the parent set has no counterpart in the real world of finite sets. It is an emergent property of infinite sets which should have told people they are entering fantasyland. This puzzled Galileo, and for good reason.

But calling it a fantasy is the way you evade points you can't argue against, so it really carries no weight.

The potential infinite makes sense, the actual infinite does not.

You think you have proven that, if a sequential algorithm is required, that all the IADs must be found before the only thing we want, the anti-diagonal. You are wrong to think so, because you deny the Axiom of Infinity, and you haven't established why it needs to be algorithmic.I have proved that all the IADs must be produced and found before ~D could be produced.

If it is not algorithmic then it is magical by definition.

No, you changed what we said into that so you can misinterpret. What we said, is that convergence applies to completely different things, and is irrelevant to finding the anti-diagonal.We have agreed that ~D is not consequent to convergence.

As far as algorithms are concerned, ~D is not the result of convergence.

No. He was talking about a completely different kind of infinity, potential infinity. Anything having to do with it is irrelevant to Cantor. Which is also the failing of everybody else you quote.Refer back to Gauss:

Gauss, Kronecker, Poincare, Weyl and others were the voices of reason. Cantor was the voice of religious delusion. All he was doing was describing a mathematical equivalent to the land of Oz. You can wander around there as much as you like, but there is no place like home [1].

[1]

ibid.Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.”

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

Since Cantor's Lemma is about injections and surjections from N to T [1], it would be interesting to hear Robert explain their meaning, particularly [1] below. Once he comprehends that, I think he will see that N and T being infinite is totally irrelevant.

[1] Cantor's Lemma could be stated as: any injection from N to T is not a surjection [2].

[2] [1] + an explicit injection [3] from N to T show that N has less cardinality than T.

[3] n 1's followed by infinite 0's suffices as an injection.

[1] Cantor's Lemma could be stated as: any injection from N to T is not a surjection [2].

[2] [1] + an explicit injection [3] from N to T show that N has less cardinality than T.

[3] n 1's followed by infinite 0's suffices as an injection.

- Gofer
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### Re: Cantor Diagonal Argument disproof

Which is exactly why a new mathematics, that accepts the Axiom of Infinity, is required. One that doesn't refer to how "large" a set is, but to a new concept where this can be true.robert 46 wrote:That a proper subset can be as large (equinumerous) as the parent set has no counterpart in the real world of finite sets.

Once it is, this property can be derived.

If you keep denying the Axiom of Infinity, yes it will seem that way. The fault is in your denial, not the mathematics.The potential infinite makes sense, the actual infinite does not.

Why? This is pure fiction on your part. And you must realize it, since you will never defend the assertion.If it is not algorithmic then it is magical by definition.

... all denied the Axiom of infinity.Gauss, Kronecker, Poincare, Weyl and others

And this is your form of posturing, believing you are superior to someone with religious beliefs. It has no place in any discussion.Cantor was the voice of religious delusion.

I agree. That doesn't mean it is invalid, just at a higher level.Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.”

- JeffJo
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### Re: Cantor Diagonal Argument disproof

JeffJo wrote:Which is exactly why a new mathematics, that accepts the Axiom of Infinity, is required. One that doesn't refer to how "large" a set is, but to a new concept where this can be true.robert 46 wrote:That a proper subset can be as large (equinumerous) as the parent set has no counterpart in the real world of finite sets.

Once it is, this property can be derived.

Well, if you make up your own rules you can prove whatever you want. That is generally how religions do it.

If you keep denying the Axiom of Infinity, yes it will seem that way. The fault is in your denial, not the mathematics.The potential infinite makes sense, the actual infinite does not.

The fault is in axioms which have no connection with reality. Baum had no trouble defining the Land of Oz either.

Why? This is pure fiction on your part. And you must realize it, since you will never defend the assertion.If it is not algorithmic then it is magical by definition.

Claiming that an infinity of characters can be selected and negated all in parallel is algorithmic??? It's like being plunked down in the Land of Oz by a tornado. See official map here:

https://en.wikipedia.org/wiki/Land_of_Oz

... all denied the Axiom of infinity.Gauss, Kronecker, Poincare, Weyl and others [were the voices of reason.]

Good for them. Hooray! for the home team.

And this is your form of posturing, believing you are superior to someone with religious beliefs. It has no place in any discussion.Cantor was the voice of religious delusion.

It is not a comparison between me and them, but between rational thought and irrational thought. I have pointed out the religious leanings of Cantor, and how he injected them into mathematics through the religious method of exaggeration.

I agree. That doesn't mean it is invalid, just at a higher level.Mathematician Solomon Feferman has referred to Cantor's theories as “simply not relevant to everyday mathematics.”

The level of pure fantasy. Is this the proper trend for mathematics???

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

Since Robert ignored my challenge, I can only conclude he doesn't know how to answer it, and isn't aware of what Cantor's theorem is really about; and so is Jeff, would it seem.

Cantor's theorem is about whether there exists some method/algorithm/schema, or whatever, pairing each member in N with each member in T, and is, in fact, a more generalized way of counting, which works regardless of the sets being infinite or not. If you can pair every member of both sets, they are equally big. If you can't, one set is bigger.

So Robert's arguing of N and T being infinite is really just a straw man.

Cantor's theorem is about whether there exists some method/algorithm/schema, or whatever, pairing each member in N with each member in T, and is, in fact, a more generalized way of counting, which works regardless of the sets being infinite or not. If you can pair every member of both sets, they are equally big. If you can't, one set is bigger.

So Robert's arguing of N and T being infinite is really just a straw man.

- Gofer
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### Re: Cantor Diagonal Argument disproof

Gofer wrote:Cantor's theorem is about whether there exists some method/algorithm/schema, or whatever, pairing each member in N with each member in T, and is, in fact, a more generalized way of counting, which works regardless of the sets being infinite or not. If you can pair every member of both sets, they are equally big. If you can't, one set is bigger.

So Robert's arguing of N and T being infinite is really just a straw man.

Remember, infinite is just a synonym for endless. It is impossible for one endless set to be bigger than another endless set. How could one possibly exceed the other??? Cantor's argument is faulty. One cannot logically stretch the imagination far enough to imply ~D exists. It is what one might call an unbelievable exaggeration- a tall tale.

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

That depends on the definition of "bigger", doesn't it?It is impossible for one endless set to be bigger than another endless set.

Fact of the matter is:

1. as I just stated, the definition of cardinality works for ANY set, finite or infinite, but counting the usual way only works for finite sets; so cardinality is more general.

2. Cantor's Lemma is about cardinality. If you are not arguing that, you are misrepresenting Cantor.

3. Cantor can come up with any definitions he likes, create a theory around it, and prove theorems with it. So if you postulate he can't do that, or that his theory isn't "physical", or real, you are misrepresenting Cantor.

4. Cantor's Lemma is NOT about whether an algorithm simulated on a computer will ever completely calculate ~D. If you are arguing that, you are misrepresenting Cantor.

So I'm saying it again: the consequence of Cantor's Lemma is that there doesn't exist a way of pairing every member of N with every member of T, be it through algorithms, quantum computers, or whatnot. That is a real, physical statement, if you will, about the world we live in. N and T being infinite is just trapdoor for the unwary.

- Gofer
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