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April 11th, 2016, 01:59 PM   #1
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Cantor's Diagonalization Refuted Simply, Simplest Version

The last thread got long and the simplest version came late in the thread. I propose it again here.

I am going to use unitary numbering. Cantor used binary numbering, which left the impression he was accomplishing something. In unitary, it is manifest he is not. Unitary uses one symbol, 1. It has equivalent expressive power to any other numbering systems. For example, 12.756 is 12 1s + 756 1s/ 1000 1s. It is unwieldy to use, but it forces there to be only one ordering of the set of all sequences in which a diagonal number can be constructed. There is a one's place, another one's place, and another one's place, so the diagonal has to proceed by using the next symbol. The lack of a symbol to the right can be interpreted as zeros if you want, that is done in unitary. But 1 1 is not a number where there is a lack of a digit between two digits, that would be the same as 11 or 2 in decimal.

The proof:

A natural number n always has a successor n+1 which is a natural number.

Unitary numbering shows a diagonal number is the equivalent of n+1.

11
111
1111
11111
111111
...

The diagonal argument cannot prove an element is not in the set of all natural numbers because it uses the definition of natural numbers to create the diagonal element n+1 that is not in the set of size n, but size n+1. N+1 is the successor to n, the successor to a natural number is also a natural number, so the diagonal number is a natural number.

The diagonal argument is a vacuous form of argument equivalent to using the successor function to make a number that is not a natural number. That is a contradiction in definitions. QED

Notice the diagonal number only shows it is not any element up to and including element n, but it has to be element n+1, which by definition is on the list. The diagonal number is simply the next element on the list for any natural number n.

In any other numbering system, the same thing occurs, and the buck is passed further down the list, but it is hard to see how necessary the diagonal number must be to the list. It is a tricky way to use an element that must be on the list later, passing the buck. The buck passing is manifest in unitary because it never gets further away than n+1.

There is no two kinds of infinite, or at least, diagonalization is not a method that can prove it or anything else besides Cantor was wrong. I used it like that, everything else was getting it wrong.

Last edited by skipjack; April 11th, 2016 at 10:39 PM.
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April 11th, 2016, 02:40 PM   #2
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This has nothing at all to do with the diagonal argument. It's a fool fumbling in the dark with concepts he understands nothing about. This is an incoherent circular argument in which the OP has decided "Cantor was wrong", written a few lines of poor guesswork and then jumped to the conclusion that "Cantor was wrong". My son of six years old produces better mathematical proofs than this. And some of them have the added benefit over this drivel of actually being true.

Last edited by skipjack; August 1st, 2016 at 08:55 AM.
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April 11th, 2016, 02:47 PM   #3
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You can keep doing that, but the argument is simple and irrefutable. Any set of sequences for which a diagonal number can be constructed is exactly that list. Other numbering systems are no different from using the same symbol over and over, 2 isn't necessary to binary, f isn't necessary to decimal, and every other symbol isn't necessary to unitary. All number systems are equivalent to using the same symbol over and over. All number systems count. This one does it manifestly.

It is so simple. You are confused about the nature of numbers.

If you think there is a number that is expressible in any other numbering system but not unitary, let me know.

Last edited by skipjack; August 1st, 2016 at 09:01 AM.
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April 11th, 2016, 03:06 PM   #4
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Quote:
Originally Posted by IntelligentDesire View Post
the argument is simple and irrefutable.
So irrefutable that it was refuted even before this thread was started. You claim to refute Cantor, but your work bears no resemblance to anything In Cantor's argument.

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Originally Posted by IntelligentDesire View Post
You are confused about the nature of numbers.
Says the guy who thinks there are natural numbers that are infinite. Nobody needs lessons from you on anything.

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Originally Posted by IntelligentDesire View Post
If you think there is a number that is expressible in any other numbering system but not unitary, let me know.
You can't represent any irrational number under the scheme you specified, which is quite a flaw for an attempt to disprove an argument (indirectly) about the real numbers.

You could try to disprove Cantor using Roman numerals too, but you'd fall into the same trap: your diagonal sequence wouldn't represent a number. It would just be nonsense. Some numbering systems do not have the properties required to exhibit the diagonal proof.

Last edited by skipjack; August 1st, 2016 at 09:02 AM.
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April 11th, 2016, 03:16 PM   #5
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Quote:
Originally Posted by v8archie View Post
You can't represent any irrational number under the scheme you specified, which is quite a flaw for an attempt to disprove an argument (indirectly) about the real numbers.

You could try to disprove Cantor using Roman numerals too, but you'd fall into the same trap: your diagonal sequence wouldn't represent a number. It would just be nonsense. Some numbering systems do not have the properties required to exhibit the diagonal proof.
This is not true. I will give you some Pi.

Pi can only be expressed to a certain number of the equivalent of decimal digits by any numbering system.

111+ 14159 1s/ 100000 1s (I hope you get the idea, it is too unwieldy to actually write it out you would have to count to 14159 and all the to 100000 with 1s)

Or 111. (14159 of these) 1s (The decimal point is just counting the other way.)

You can go as far as you like. That is all anything but the symbol Pi does, but numbers express it to a degree of fidelity.

The really cool part, which it should be for someone interested in number theory, is that imagining the infinite decimal expansion is exactly imagining 11111...s/ a second 1111111...s or just one dot and 111... . One kind of infinite, the counting kind, and you can't count there. The other symbols don't express anything but 1 in an infinite sequence, when you think of other symbols in an infinite sequence you are thinking of a certain number of 1s for a finite part of it, or boldly all 1s to imagine infinite representation.

Numbers. They can cycle symbols and count or just use one symbol. But numbers only count, unless you want them to mean something else. Then you can map a binary number on to 1s and you still have a machine language or whatever else.

Last edited by skipjack; August 1st, 2016 at 08:58 AM.
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April 11th, 2016, 04:50 PM   #6
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Yes, well that's obvious nonsense isn't it? Your answer is to give a rational number or a construction that is exactly the same for every irrational between 3 and 4 (try $\sqrt{11}$ it's an infinite string of 1s "followed by" another infinite string of 1s, just like $\pi$).

Last edited by skipjack; August 1st, 2016 at 08:58 AM.
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April 11th, 2016, 05:20 PM   #7
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Quote:
Originally Posted by v8archie View Post
Yes, well that's obvious nonsense isn't it? You answer is to give a rational number or a construction that is exactly the same for every irrational between 3 and 4 (try $\sqrt{11}$ it's an infinite string of 1s "followed by" another infinite string of 1s, just like $\pi$.
Have you ever seen an infinite decimal expansion yourself, or just part of one? You could say part of one is different from another, but the full thing is just infinite symbolism with an arbitrary number of symbols.

The ratio of a circumference to a radius of a circle and the square root function given a number argument are ways to get a number. I have seen the numbers in decimal form and they were finite parts with a "..." Which was the equivalent of what I did for Pi and saying but there is more to the process.

You want to see an infinite expansion, you want to see a number after a decimal point, 1s work, and if you see anything else it is just because you have gotten so used to the symbols you forgot they are just arbitrary and you can replace them with different ones or a repetition of one symbol.

You don't want to work with unitary, but it destroys Cantor's diagonalization and a lot of misconceptions about what numbers are. I think it reveals that you have some fuzzy idea about what "infinite" amounts of numbers are, they are infinite symbolism. You either get a precise part, and it come be done with any symbols, or there is the idea of the whole thing and what you do with symbols is in your head in whatever number language you are most comfortable. If you used unitary they are all 1s and that is every complete infinite sequence.

Last edited by skipjack; August 1st, 2016 at 08:59 AM.
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April 11th, 2016, 05:46 PM   #8
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Originally Posted by IntelligentDesire View Post
Have you ever seen an infinite decimal expansion yourself, or just part of one?
No, but then I've never seen Indianapolis either, but I'm pretty sure that it exists. I've never seen all of the US at one time either, but again, I'm prepared to accept that it's there.

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I have seen the numbers in decimal form and they were finite parts with a "..." Which was the equivalent of what I did for Pi and saying but there is more.
No. Because in binary, ternary, decimal, hexadecimal, etc., irrational numbers have a unique expansion, and that expansion is unique among the representation of numbers under that system. The infinite sequence of digits that represents $\pi$ under the decimal system doesn't represent any other number. But your unitary system has exactly the same expansion for every irrational number.

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1s work, and if you see anything else it is just because you have gotten so used to the symbols you forgot they are just arbitrary and you can replace them with different ones or a repetition of one symbol.
You can use any symbols you like, it doesn't bother me. But unfortunately you hadn't noticed that all irrationals share the same expansion in your system, so it's useless for those. You also failed to notice that changing a digits is vastly different from adding a symbol. Unfortunately because now I've pointed it out to you, you will refuse to accept either as truth. Your ego won't allow it.

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You don't want to work with unitary
No, because it's not powerful enough to do what you want to do with it.

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Originally Posted by IntelligentDesire View Post
but it destroys Cantor's diagonalization
No, Cantor's diagonalisation doesn't work with it because you can't perform the steps. The two are simply not compatible. Claiming that it proves Cantor wrong is like claiming that your CD player is faulty because it doesn't produce music when you put a jam sandwich in it instead of a CD.

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Originally Posted by IntelligentDesire View Post
I think it reveals that you have some fuzzy idea about what "infinite" amounts of numbers are
Again, you have amply demonstrated that you have no idea about the infinite. Sure you've got some ideas that you believe, but you only have your faith to tell you that they are true. You have no mathematics to back you up. For the most part I do - and if I need to deal with anything I'm not sure about I'll study and understand the literature before I make a final judgement.

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Originally Posted by IntelligentDesire View Post
If you used unitary they are all 1s and that is every infinite sequence.
Which is why it is not fit for purpose. Clearly not all infinite sequences are the same. Your claim to the contrary is explicitly saying that $\pi=\sqrt11$.

Nothing would please me more than for you or somebody else to come up with a genuinely interesting question about (or even challenge to) Cantor, because that could provide the basis for an interesting and informed debate from which we might learn something. All we are learning from your posts is just how stubborn and deluded somebody with no understanding can be.

Last edited by skipjack; August 1st, 2016 at 09:01 AM.
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April 11th, 2016, 06:11 PM   #9
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You have the same kind of comment for each of my comments, so let me address that. A representation of an infinite decimal expansion involves expressing infinite.

The finite part, where you can tell all the operations, which is greater than the other and that the square root of 11 is not equal to Pi, you do without ever using actual infinite symbolic representation, you use one to a degree. That can be done with unitary just like it can be done with decimal.

When you have infinite symbolism, only the finite part matters in a decimal expansion, the rest is expression of infinite magnitude. You can find every finite part the same and express it with unitary, but you can't express the whole thing with decimal and you can't express the whole thing with unitary. What are you imagining, 3s and 7s and 2s flying at you endlessly, like the wind on your face? At every point those are unitary expressions too, but the breeze looks different, like it would in hex or binary.

Think about infinite as a number, one number. The 3s, 7s, and 2s, don't mean anything, it only matters that there is an infinite count. One infinite of bigness is the same as another, but at finiteness the 3s and 7s and 2s and how many 1s there are matters. It is the same way at unbound precision smallness, you are just moving toward the end that doesn't matter at all, the infinitely small part. At infinite precision you are just expressing the exact same concept of infinite as a big number, you just pointed to smallness at the point of nothing.

In other words, all infinite decimal expressions point to the magnitude of nothing, just as infinite bigness is the same. If you didn't use them all, you are just working it out to a finer and finer point and the symbols matter, when you use them all you are expressing precision at the point of nothing.

That is how I would put it, and it is consistent with the view of numbers I got when I grabbed for unitary numbering and saw a lot more about numbers, which occurred during the course of our argument in trying to get the simplest refutation of Cantor. Infinite is clear now to me, and that is more beautiful than multiple kinds of fuzziness. The fuzziness, the I can't do it, hurt, and this proof is relief.

Last edited by IntelligentDesire; April 11th, 2016 at 06:54 PM.
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April 11th, 2016, 07:46 PM   #10
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What you fail to realise is that $\pi$ has an infinite expansion that is unique to it. You will never see it all, but it is still there, and that is the expansion that is used in the Cantor proof (in the form of a sequence, rather than a number) because Cantor wasn't writing about rational approximations, he was writing about the real numbers. You can tell because his proof sites infinite sequences of digits, not finite sequences of digits. (By the way, ${22 \over 7}$ is also the same as $\sqrt11$ in your unitary scheme.

But again, the biggest problem with unitary is that you can't perform one of the steps in the process. In your description, you made up something else to do instead, but that doesn't make your process the same as Cantor's, it's different because you do a different thing (add 1 instead of change a digit).
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