malcolmmcswain
  • malcolmmcswain
Cantor's Diagonal Proof (Tutorial)
Mathematics
  • Stacey Warren - Expert brainly.com
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
malcolmmcswain
  • malcolmmcswain
What is Cantor's Diagonal Proof? Georg Cantor was a brilliant mathematician who invented all sorts of useful tools for categorizing the endlessly confusing thing called infinity. He even went so far as to say that there were different sizes of infinity. He said that the first kind of infinity was Aleph Null. \[\aleph _{0}\]
malcolmmcswain
  • malcolmmcswain
Aleph Null was the countable infinity. Cantor said the natural numbers, (0,1,2,3,4,5…) the integers, (0,-1,1,-2,2-3,3…) and even the rational numbers (1/1,-1/2,2/3…) were all countable. However, he said the REALS (a number set seemingly to the rationals, but including the pesky numbers like pi called irrational) were NOT COUNTABLE.
malcolmmcswain
  • malcolmmcswain
Well, of course he needed to prove this, so this is what he came up with. Suppose, for instance, you WERE able to make a list of every single real number. This infinitely long list could be considered countable, right? Given an infinite amount of time, you should be able to list out every single real… Wrong. Let's take a look at this list again… |dw:1450303496619:dw|

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malcolmmcswain
  • malcolmmcswain
There is a way to build a number that will not be on this list. If I take the first digit of the first number, and add one, then take the second digit of the second number and add one, then take the third digit of the third number, and add one (and so on forever) we build an infinite, non-repeating decimal that won't be on the list. |dw:1450303689785:dw|
malcolmmcswain
  • malcolmmcswain
How? Well, it can't be the same as the first number, because the first digit of the created number is one more than the first digit of the first number on the list, and the second digit of the created number is one more than the second digit of the second number on the list, and so on… Therefore, the reals are uncountable.
malcolmmcswain
  • malcolmmcswain
Cantor named this new kind of infinity Aleph 1, \[\aleph _{1}\] and his discovery revolutionized the way we think about infinity.
malcolmmcswain
  • malcolmmcswain
Thank you for reading. :)
Danisaur
  • Danisaur
*claps* e_e
malcolmmcswain
  • malcolmmcswain
Lol. :)
AlexandervonHumboldt2
  • AlexandervonHumboldt2
=]
AlexandervonHumboldt2
  • AlexandervonHumboldt2
*exploding forever*
malcolmmcswain
  • malcolmmcswain
http://prntscr.com/9f1p95 my tutorial that I spent hours working on gets less medals than someone who pretends to clap v_v lol that's openstudy I guess. :\
AlexandervonHumboldt2
  • AlexandervonHumboldt2
SOTP THIS MADNESS! WHY FOR CLAPS ANHYONE GET 100000 MEDALS, AND THE ONE WHO MADE A TUTORIALS GET -556465??????????
anonymous
  • anonymous
nice tutorial :)
malcolmmcswain
  • malcolmmcswain
Thanks. That really means a lot, @OlegPetrovich_PR =']
anonymous
  • anonymous
Best tutorial ever :)
AlexandervonHumboldt2
  • AlexandervonHumboldt2
woooooooooooooooo ==]
malcolmmcswain
  • malcolmmcswain
Thank you guys so much. (Shurik's alts) Shurik, you're the only one who appreciates the work I put into answering questions and writing tutorials ;)
AlexandervonHumboldt2
  • AlexandervonHumboldt2
LOL
AlexandervonHumboldt2
  • AlexandervonHumboldt2
you better not write that they are ma alts lol
AlexandervonHumboldt2
  • AlexandervonHumboldt2
@OrangeMaster =]
anonymous
  • anonymous
Well done. :)
AlexandervonHumboldt2
  • AlexandervonHumboldt2
yay
amistre64
  • amistre64
if i were to have any issue with Cantors proof, it is that he attempted to use a numerical representation for his irrational numbers. irrational numbers are presented as symbols: \(\pi,e, \sqrt{n},L,\) etc ... we can make this infinite set countable simply by attributing each symbolic representation to an element of the set: {\(x_1,x_2,x_3,...\)}. All I really see is that Cantor proved the set of irrationals to not be able to be organized in a reliable manner if we are to represent them as a decimal expansion.
ganeshie8
  • ganeshie8
I think Cantor proved more than that. He showed that a bijection doesn't exist between the set of natural numbers and the set of real numbers. Since a bijection doesn't exist, those two infinite sets cannot have the same cardinality
malcolmmcswain
  • malcolmmcswain
@ganeshie8 Ah, yes, Cantor's continuum hypothesis. I should have included that. The continuum hypothesis, I believe, also proves that there are infinite alephs, each larger than the last.
ganeshie8
  • ganeshie8
my reply is for @amistre64 :)
ganeshie8
  • ganeshie8
I liked your tutorial... it is a nice read :D
malcolmmcswain
  • malcolmmcswain
Haha, I know. I was remarking that I should have included it in my tutorial. Also, thank you very much. :)
Willie579
  • Willie579
Amazing tutorial! :D I don't think I get it because I'm only in Middle School. :P
malcolmmcswain
  • malcolmmcswain
I learned about Cantor's diagonal proof when I was in middle school. :P However, it's a tough concept to grasp lol.
amistre64
  • amistre64
The proof is that you create a sting of digits, whos kth digit for the kth does not match the kth decimal expansion ... since the sting of digits that was formed is spose to be in the set to start with, but you cannot place it anywhere in the set, we have a contradiction. the contradiction is that the set contains every string of digits possible and we created a string of digits that we can never find a place for. Now as far as cardinality goes, if the irrationals are more dense then rationals, shouldnt we be able to find an interval between 2 irrationals that does not contain at least 1 rational?
amistre64
  • amistre64
just some thinking i do when i cant sleep :) spose we construct 2 distinct irrational numbers as close as possible. to start out, let them have the exact same decimal expansion for however long we want. at some point they have to differ - otherwise they would be the same number - but we want to make it so that we cannot squeeze a rational value inbetween them. a=.1415... b=.1414... the smallest difference possible between the 50s and 40s is: 50 and 49, so the only hope we have is to make the next selection 0 ad 9. a=.14150... b=.14149... in order to keep these irrational, we cannot keep repeating 0 and 9, one of them eventually has to change. the smallest possible change in our 499 is to 498, at which time we can determine a rational number between them. a=.141500... n=.141499 b=.141498... so we have at least one rational between them, but we also have to eventually change the other irrational as well, the smallest change from 500 is 501, at which time we can determine a second rational number between them. a=.1415001... k=.1415000 n=.1414990 b=.1414988... so, there exists at least 2 rational numbers between the smallest possible interval of 2 irrational numbers; yet we know that the set of rational numbers are countable.
amistre64
  • amistre64
Cantor's proof assumes that the Reals can be formed into a countable list (using a decimal expansion as the basis for that list). And he proves that the assumption is flawed ... there is no good way to organize the list using a decimal expansion as the means of representing the Reals.
anonymous
  • anonymous
See this ted talk... it addresses all (most) of this (sorry, didn't bother to read all of the posts). http://ed.ted.com/lessons/how-big-is-infinity I show this to my students. it's fairly short < 7 mins. and get them thinking...
amistre64
  • amistre64
the part from 1:50 to 2:10 is making my point. There are ways to try to match things up that simply dont work out for us, and "it doesnt matter if you try to match and it doesnt work ... as long as you can find some way".
anonymous
  • anonymous
http://www.ams.org/samplings/feature-column/fcarc-irrational4
AloneS
  • AloneS
I love this one actually too!
anonymous
  • anonymous
Good tutorial, although that video pgpilot shared was much easier to understand for an audio learner like me
anonymous
  • anonymous
People ought to also check out Continued Fractions. All rationals can be expressed as finite continued fractions while the irrationals cannot. Here is a link to wikipedia but you can look at wolfram's or another page if you don't trust wikipedia. https://en.wikipedia.org/wiki/Continued_fraction
ikram002p
  • ikram002p
dang!! pretty nice.
Astrophysics
  • Astrophysics
Nice tutorial, thanks!
jabez177
  • jabez177
Excellent job, @MalcolmMcSwain! :D

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