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Don Stockbauer
Charter Member

Oct2000, 12:18 PM (EST) 

"Cantor's diagonal method"

I know there has to be some simple error in the following "proof" that the cardinality of the integers and reals is the same, but I cannot easily identify it. After the Cantor diagonal real is formed(which is nowhere in the table), merely insert it into the table and assign the next higher unused integer index to it. The argument is that there will always be another integer available, thus a onetoone mapping can be formed. What is the error in this?



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alexb
Charter Member
2467 posts 
Oct2000, 12:25 PM (EST) 

1. "RE: Cantor's diagonal method"
In response to message #0

> After the Cantor diagonal real is formed > (which is nowhere in the table), merely > insert it into the table and assign the > next higher unused integer index to it. > The argument is that there will always > be another integer available, thus a > onetoone mapping can be formed. What > is the error in this?Well it's more a question of explaining this nonsense than refuting it. The above is just a collection of words that together make no sense.
 "assign the next higher unused integer"
In the diagonal process you used up all available integers to start with.  "thus a onetoone mapping can be formed"
The word "thus" is used in mathematics quite frequently. This does not mean that what follows after it is always true. Even assuming there is always the "next higher unused integer", it does not follow that what is left of the reals and what is left of integers (?) stand in a 11 correspondence. 


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Don Stockbauer (Guest)
guest

Nov2100, 01:49 AM (EST) 

2. "RE: Cantor's diagonal method"
In response to message #0

I cannot claim the following as good mathematics; it's merely a vision I keep having. Cantor's infinity of infinities is a reflection of his troubled mind. Suppose we postulate that there is only one infinity, that of the counting numbers. All others are merely variants of it. Let us further propose that even this infinity has no business in the real world. Godel's Incompleteness Theorem falls, allowing for consistent and complete complex systems; in particular, the General System, the Universe, is freed from inconsistencies. As we expand into it we do not face the Chaos that inconsistency brings. The application of Systems Theory to the General System will advance Humanity far beyond where we can even dream at this point (see Arthur C. Clarke’s Three Laws). Time will tell. Don Stockbauer Donstockbauer@hotmail.com



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alexb
Charter Member
2467 posts 
Nov2600, 00:58 AM (EST) 

3. "RE: Cantor's diagonal method"
In response to message #2

>I cannot claim the following as >good mathematics; I have no problem with this. >Cantor's infinity of infinities is a >reflection of his troubled mind. May be. It does not mean that anything is wrong with his mathematics. I knew a fellow who was only able to generate mathematics when drunk. But the mathematics was superb. > Suppose we postulate that >there is only one infinity, >that of the counting numbers. I do not understand how you can postulate that. Can you postulate that there are no cats? dogs? parrots? that I am unmarried? >Let >us further propose that even >this infinity has no business >in the real world. May be it would be wiser just to look around? >Godel's Incompleteness Theorem falls, Are you sure Godel ever mentioned Cantor's name? I recommend a popular exposition by E. Nagel and J. R. Newman "Godel's Proof", New York University Press, 1986 > allowing >for consistent and complete complex >systems; in particular, the General >System, the Universe, is freed >from inconsistencies. As we >expand into it we do >not face the Chaos that >inconsistency brings. Are you sure inconsistency or Chaos are such a drag on Humanity's advance? I am confident it's the other way around. > The application >of Systems Theory to the >General System will advance Humanity >far beyond where we can >even dream at this point >(see Arthur C. Clarke’s Three >Laws). Time will tell. Sure. Alexander Bogomolny



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Don Stockbauer (Guest)
guest

Jun1301, 02:38 PM (EST) 

4. "RE: Cantor's diagonal method, yet again"
In response to message #3

>Cantor's infinity of infinities is a >reflection of his troubled mind. May be. It does not mean that anything is wrong with his mathematics. I knew a fellow who was only able to generate mathematics when drunk. But the mathematics was superb. Sounds unlikely. Do you know a lot of drunks? > Suppose we postulate that >there is only one infinity, >that of the counting numbers. I do not understand how you can postulate that. Can you postulate that there are no cats? dogs? parrots? that I am unmarried? Cats, dogs, parrots; all members of the real world. Cybernetics makes clear that infinity is not. But mathematicians are allowed to play with it and produce their paradoxes. >Let >us further propose that even >this infinity has no business >in the real world. May be it would be wiser just to look around? Let me know when you find it. >Godel's Incompleteness Theorem falls, Are you sure Godel ever mentioned Cantor's name? I recommend a popular exposition by E. Nagel and J. R. Newman "Godel's Proof", New York University Press, 1986 My mistake. He actually imported the Epimenides paradox into formal number theory. A system which folds back in on itself is said to be "diagonalized", thus the confusion. Appypollyloggies. > allowing >for consistent and complete complex >systems; in particular, the General >System, the Universe, is freed >from inconsistencies. As we >expand into it we do >not face the Chaos that >inconsistency brings. Are you sure inconsistency or Chaos are such a drag on Humanity's advance? I am confident it's the other way around. Again you're right. We need both order and chaos as complementary concepts to keep the brew stirred. Mistake # 2. > The application >of Systems Theory to the >General System will advance Humanity >far beyond where we can >even dream at this point >(see Arthur C. Clarke’s Three >Laws). Time will tell. Sure. I'm glad you agree with me. Every day it keeps getting clearer and clearer to me. Have a nice day. 


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alexb
Charter Member
2467 posts 
Jun1301, 03:00 PM (EST) 

5. "RE: Cantor's diagonal method, yet again"
In response to message #4

>>Cantor's infinity of infinities is a >>reflection of his troubled mind. > >May be. It does not mean >that anything is wrong with >his mathematics. I knew a >fellow who was only able >to generate mathematics when drunk. >But the mathematics was superb. > > >Sounds unlikely.What sounds unlikely? That I knew the fellow? Do you dare question my sincerety? What caddishness! > Do you know >a lot of drunks? Why? What is the nature of your interest? Are you looking for drunks to make their acquaintance? >> Suppose we postulate that >>there is only one infinity, >>that of the counting numbers. > >I do not understand how you >can postulate that. Can you >postulate that there are no >cats? dogs? parrots? that I >am unmarried? > >Cats, dogs, parrots; all members of >the real world. Cybernetics >makes clear that infinity is >not. But mathematicians are >allowed to play with it >and produce their paradoxes. Nope. You can't postulate anything you want. The system must be consistent. Existence of multiple infinities is a consequence of other postulates. This is where you should start if you want to eliminate the multitude. >>Let >>us further propose that even >>this infinity has no business >>in the real world. > >May be it would be wiser >just to look around? > >Let me know when you find >it. You can't sit on two horses. Either you deal with the abstract or with the concrete. Do you know what the real world is? Is Zeno's paradox about the real world? When I suggested to look around, I meant to think about what you see, not about what is (or may be) out there. >>Godel's Incompleteness Theorem falls, >Are you sure Godel ever mentioned >Cantor's name? I recommend a >popular exposition by E. Nagel >and J. R. Newman "Godel's >Proof", New York University Press, >1986 > >My mistake. He actually imported >the Epimenides paradox into formal >number theory. A system >which folds back in on >itself is said to be >"diagonalized", thus the confusion. >Appypollyloggies. > >> allowing >>for consistent and complete complex >>systems; in particular, the General >>System, the Universe, is freed >>from inconsistencies. As we >>expand into it we do >>not face the Chaos that >>inconsistency brings. > >Are you sure inconsistency or Chaos >are such a drag on >Humanity's advance? I am confident >it's the other way around. > > >Again you're right. We need >both order and chaos as >complementary concepts to keep the >brew stirred. Mistake # >2. > >> The application >>of Systems Theory to the >>General System will advance Humanity >>far beyond where we can >>even dream at this point >>(see Arthur C. Clarke’s Three >>Laws). Time will tell. > >Sure. > >I'm glad you agree with me. > Every day it keeps >getting clearer and clearer to >me. Bless you. You are lucky. >Have a nice >day. You too.



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Don Stockbauer (Guest)
guest

Jun1801, 07:18 AM (EST) 

6. "Principia Cybernetica"
In response to message #5

I have lately become enamored of the Principia Cybernetica web: https://pespmc1.vub.ac.be An almost frightening concept there, the uniting of all human knowledge; science, math, ... In fact, they all become subdisiplines to be hung on the framework of cybernetics. As one peruses the system, one becomes sucked into it, realizing that all principles being studied apply to the observer, selfreference occurs, feedback loops are created, one's mind clears of a lot of conflicting junk. I highly recommend it to your readers. Maybe there's hope for the species after all. Perhaps Arthur C. Clarke's three laws will be allowed to manifest themselves, having a substrate to arise from rather than a cold dead planet. Don Stockbauer donstockbauer@hotmail.com 


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Don Stockbauer
guest

Jan2410, 11:01 PM (EST) 

7. "RE: Principia Cybernetica"
In response to message #6

The answer to this whole thread on Cantor's Diagonal Theorem is that unless one wants to spend an infinite amount of time debating infinity, go with the 2 forms of it from a cybernetic (ie, useful) viewpoint: 1. Potential 2. Actualized (Nice intervening 9 years, all. Much has happened). 


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