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Cantor's Set of Countable infinite and Uncountable infinite Infinites

You may know and you may have proved that Set of Real Numbers Between 0 and 1 are Uncountably Infinite. Mean we Can not Map Every number of that set on a different Natural Number.

I got a Technique by which I would be able to Map all Real numbers between 0 and 1 on a different Natural Number. Technique is Simple Replace the Decimal Point with 1 and Map the Original on that Number Such that Map 0.0003 on 10003 and 0.03 on 103

By using this Technique we Would be able to Map all Real Numbers Between 0 and 1 on Natural Numbers. And All of those Natural Numbers will be starting with 1 so we will be having other Numbers as well on which No Number will be mapped like 2 or 211 or 79 So This Means Set of Natural Numbers is Grater then Real Numbers Between 0 and 1. So Set of Real Numbers Between 0 and 1 is Countably Infinite.

What's Ur Opinion ?

David Robinson
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Ayub Khan
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    This question appears to be off-topic because it is about math and has nothing to do with programming – David Robinson Oct 08 '13 at 17:29
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    Incidentally, your logic is incorrect since real numbers can go on infinitely after the decimal point. Take pi-3, which is a real number (.14159265...) between 0 and 1. If you replace the decimal point with a 1 (114159265...), it will be infinitely large, and therefore not a natural number. – David Robinson Oct 08 '13 at 17:31
  • Good Point , that was one of point due to which I share this here. My Argument is we Get a irrational number sequence when we take Under-root(2) or pi but when we are talking about numbers between 0 and 1 we are talking about numbers which are not infinite , because those are only created when use some mathematic operation on different numbers. – Ayub Khan Oct 08 '13 at 17:37
  • What? No. You said "real numbers between 0 and 1." This includes numbers that go on infinitely: that's how [real number](http://en.wikipedia.org/wiki/Real_number) is defined. If you say "only those that don't go on infinitely after the decimal point," you are not talking about real numbers anymore. – David Robinson Oct 08 '13 at 17:39
  • You are saying if there is a number which is having infinite length is not going to be Natural Number. this number is not going to end . but assume if there is some end then if we replace . with 1 it would be Natural number of infinite length – Ayub Khan Oct 08 '13 at 17:45
  • `but assume if there is some end` a) there is no end, plain and simple. The number doesn't even have to be irrational for your approach to break: take 1/3, which is a rational number between 0 and 1: it looks like `.333333...`. Replace the decimal with a 1 and you get `1333333...` a number of infinite length – David Robinson Oct 08 '13 at 17:47
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    Furthermore, as soon as you say "it's a natural number *of infinite length*" you are conceding the point. There is no such thing as a natural number of infinite length. And if you redefine natural numbers to allow them to be of infinite length, then they *are no longer countable* – David Robinson Oct 08 '13 at 17:48
  • We are talking about infinites. When we map numbers we simply give a method by which it could be mapped, if some set is Countable in it there would be a very large number uncountable for us but we give argument that it would be mapped using technique. – Ayub Khan Oct 08 '13 at 18:02
  • Same thing here there could be very larger numbers but we say at some point they would be mapped. So if is number is of Infinite length is just a infinity for us , Both are infinites the number on which its going to be mapped or the Real number we are mapping. – Ayub Khan Oct 08 '13 at 18:02
  • No, because there are uncountably many "infinite numbers." That is *different* than there being countably infinite natural numbers. Therefore, your mapping is *not* onto the set of natural numbers, which is what you were trying to do. – David Robinson Oct 08 '13 at 18:06
  • Here's another way to put it: how many real numbers are there when you *don't* constrain it to be between 0 and 1? Well, we can take any real number and replace the decimal point with a 1. Now we have some number of digits to the left of that 1, and then we have possibly infinitely many digits to the right of that 1. We've just mapped it to *the exact same set* that you did in your example: an infinitely long number can't get "longer." So did you just prove that the set of all reals is countably infinite? – David Robinson Oct 08 '13 at 18:09
  • There is another point we are mapping between 0 and 1, those irrational numbers are only produced when we do some mathematics like under-root or division. without those mathematic operation there is no number which is never going to end mean having infinite size, so This way it's countable set. – Ayub Khan Oct 08 '13 at 18:20
  • Give a single example of any Real Number between 0 and 1 with out any Mathematic calculations which has a infinite size and I will take back my argument that this is a countable set. – Ayub Khan Oct 08 '13 at 18:21
  • Why do you get to specify "without any mathematic calculation"? Do you or do you not admit that the square root of 2 minus 1 is a real number between 0 and 1? – David Robinson Oct 08 '13 at 18:26
  • Or how about 1/3? Is it your opinion that 1/3 is not a real number? Or is it your opinion that 1/3 is not between 0 and 1? – David Robinson Oct 08 '13 at 18:29
  • ok if we start counting form 0 to 1 will we get those number in our sequence. – Ayub Khan Oct 08 '13 at 18:36
  • if you're saying "I am defining the real numbers between 0 and 1 as those we would get by counting between 0 and 1," then you are *by definition* creating a countable set. But it has no resemblance to the set of real numbers as everyone else defines it. – David Robinson Oct 08 '13 at 18:37
  • ok leave the counting just tell me if we are having a set of all real Numbers Between 0 and 1 will there be a entry which will be having infinite length. – Ayub Khan Oct 09 '13 at 02:11
  • While we Define sets as Collection of well define and distinct objects. So if a number is Never going to end is it well define? – Ayub Khan Oct 09 '13 at 02:17
  • `will there be a entry which will be having infinite length` Yes. For example, pi-3, 1/3, sqrt(2)-1, and so on. Please read up on [real numbers](http://en.wikipedia.org/wiki/Real_number): `Any real number can be determined by a possibly infinite decimal representation such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one` – David Robinson Oct 09 '13 at 02:26
  • As for whether they are "well defined": some infinite numbers are rational (like 1/3) and so are very simple to define (the ratio of 1 and 3). Some infinite numbers are irrational (like sqrt(2)-1), some are [transcendental](http://en.wikipedia.org/wiki/Transcendental_number). That doesn't mean they aren't "well defined." – David Robinson Oct 09 '13 at 02:30
  • So why cant we Map 0.333.... on 1333.... ? – Ayub Khan Oct 09 '13 at 04:53
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    What is 1333...+1? An integer has to have a well defined and unique successor, so 1333... is not an integer. – Joni Oct 09 '13 at 06:31
  • Can we get this Number by Moving from 0 to 1 ! – Ayub Khan Oct 09 '13 at 09:41
  • Just Give me a proof of any Irrational No between 0 and 1 with out using any Mathematic Operation . I will accept your argument and will accept that this set is Uncountable. – Ayub Khan Oct 09 '13 at 09:56
  • `with out using any Mathematic Operation` You're asking a question about math. Saying "you can't use any math in the answer" is nonsense. – David Robinson Oct 09 '13 at 12:38
  • `Can we get this Number by Moving from 0 to 1` a) Is 1/3 a real number? b) Is 1/3 greater than 0? b) Is 1/3 less than 1? If all of these are true, than 1/3 is a real number between 0 and 1. Which of those statements do you disagree with? – David Robinson Oct 09 '13 at 12:45
  • I do agree with all of those Statements. But My Question is why we are using Mathematics Operations to define a number? – Ayub Khan Oct 09 '13 at 13:55
  • @AyubKhan: Because 1/3, sqrt(2), and pi cannot be written as a decimal form: they would be infinitely long. If you agree that 1/3 is a real number, how would you prefer me to write it down? – David Robinson Oct 09 '13 at 14:54
  • Examples of concrete constructions of irrational numbers: Consider a square. Its diagonal is longer than its side, but less than twice as long. By how much longer is it? Consider a triangle with sides of equal length. How tall is it? What's its area? Consider a circle within a square. How much of the area of the square is contained within the circle? – Joni Oct 09 '13 at 14:58
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    I'm voting to close this question as off-topic because it's about math and not computer science – Rune FS Apr 06 '17 at 21:25
  • I'm voting to close this question as off-topic because it is about [math.se] instead of programming or software development. – Pang Apr 08 '17 at 05:09
  • I'm voting to close this question as off-topic because it is about mathematics and not programming. – Mark Rucker Aug 12 '17 at 00:12
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    I'm voting to close this question as off-topic because it's about mathematics. – Bob__ Oct 24 '17 at 12:30
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    I'm voting to close this question as off-topic because this is about mathematics without anything given that ties it to programming. – David Young Aug 21 '19 at 01:36

2 Answers2

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The set of real numbers between 0 and 1 is uncountably infinite, as shown by Cantor's diagonal argument which you are familiar with.

What may be surprising to you is that the set of rational numbers between 0 and 1 is countably infinite. That is, there is a 1-to-1 correspondence between the integers and all fractions and numbers with a finite decimal expansion. You can find the proof here.

Joni
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  • @AyubKhan: What flaws would those be? – David Robinson Oct 09 '13 at 02:39
  • Power Set of Natural Numbers surely will be greater then the Natural Numbers so we need More Natural Numbers to map them on them. – Ayub Khan Oct 09 '13 at 04:55
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    In diagonal argument we define them till some point mean till that if we will be taking diagonal it will not be there but if we keep on moving we will get that diagonal. this will continue. – Ayub Khan Oct 09 '13 at 04:57
  • There could be other ways to map those Numbers on Natural Numbers. Will discuss that after real numbers Problem. im using a different method to map them on natural numbers – Ayub Khan Oct 09 '13 at 04:59
  • The technique you mention in the question is wrong as already pointed out the in comments. At best you have shown an equivalence between the reals and the p-adic integers. – Joni Oct 09 '13 at 06:27
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This doesn't work because an arbitrary non-rational real number such as 0.5123129421... is a legitimate real number but the number 15123129421... isn't. in the case of the former, you can point out (at least in principle) where along the number line it would lie, but for the latter, it's impossible. Try to say out 15123129421... as one number (like 1022 is one thousand and twenty two). You won't be able to, because such number is not a natural number.

methmatician
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