October 17th, 2016, 04:00 AM  #81  
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1383  Quote:
As there is no "list at infinity", that doesn't imply that the reals are countable. Suppose instead that you make a list of n identical sequences, all of length 10, each sequence being 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. You can choose a single appropriate digit from these sequences so that your chosen digits are 7, 0, 7, 1, 0, 6, 7, . . . continuing until n digits have been listed. By continuing in this fashion (with increasing values of n), your choices can form the first n places of the decimal expansion of the square root of 1/2, or any other real in [0,1). You need a countable number of steps to do this, so by your logic you could claim that the reals in [0,1) are countable. That would lack proper justification, as being able to take an arbitrarily large, but finite, number of steps towards some goal doesn't imply that the goal is itself countable (in any sense). Even if you took an infinite sequence of steps towards the goal, that still wouldn't prove that the goal is countable.  
October 17th, 2016, 06:14 AM  #82  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,134 Thanks: 88  Quote:
There is no list of real numbers at infinity (a number), but you can talk about an infinite (unending) list. The list of nplace rational numbers gets closer and closer to the list of real numbers and remains unique as n gets larger and larger. That's the fundamental concept of real analysis. If a property is true for all (any) n, in this case uniqueness and countability, it is true for countably infinite n. As for the second part of previous post, I find the example irrelevant and the conclusion unintelligible. Last edited by zylo; October 17th, 2016 at 06:23 AM.  
October 17th, 2016, 06:48 AM  #83  
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,049 Thanks: 680 Math Focus: Physics, mathematical modelling, numerical and computational solutions  Quote:
Quote:
Also... your scheme to list all of the real numbers doesn't contain all of the real numbers (e.g. 1/3 is not in your list). It can only contain that number if n is allowed to be infinity (not just approach it), but you already objected to that. Last edited by Benit13; October 17th, 2016 at 06:59 AM.  
October 17th, 2016, 07:57 AM  #84  
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1383  Quote:
That doesn't imply the countability of the irrationals, as all your countable lists contain only decimals corresponding to rational values. You can use a selection process to find listed rationals that approach any irrational real number in [0,1), but that doesn't show that those reals are countable, because each finite selection of rationals gets used for such an approach to infinitely many reals (that you haven't shown are countable).  
October 17th, 2016, 08:45 AM  #85  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,938 Thanks: 2265 Math Focus: Mainly analysis and algebra  Quote:
Moreover, when $n$ gets larger and larger, there is no suggestion in real analysis that this has anything to do with "infinity". It just means that $n$ can become arbitrarily large. Related to your discussion, this just means that your $n$place decimals can become arbitrarily long, but by virtue of having a length they remain finite. Quote:
The concept of limit in real analysis applies to the values of the elements of a sequence. In particular, the sequence converges to the value $L$ only if for any (arbitrarily small) value $\epsilon$ we can always find an $N$ such that the value of every element of the sequence after the $N$th is within $\epsilon$ of $L$. At no point does any concept of "infinity" get involved in this definition. You cannot use this to to tell you anything about set theoretic infinities. Last edited by v8archie; October 17th, 2016 at 08:50 AM.  
October 19th, 2016, 02:49 PM  #86 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,134 Thanks: 88 
My posts, buried in this thread, are summarized and generalized, simply and clearly, in: Decimal Representation and Expansion 
October 19th, 2016, 03:09 PM  #87 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,938 Thanks: 2265 Math Focus: Mainly analysis and algebra 
So the sum total of your posts here is that you believe, on no evidence whatsoever, that whatever is true for finite cases must also be true for infinite cases. Moreover, you prefer to ignore the evidence presented to the contrary, instead repeating your bogus claim as if nobody had commented at all. It doesn't matter how many times you repeat nonsense, it remains nonsense. 
October 19th, 2016, 08:55 PM  #88 
Global Moderator Joined: Dec 2006 Posts: 17,919 Thanks: 1383  The thread you linked to doesn't give any mathematical argument at all; it does little more than define a term that you haven't used in any of your posts here. In particular, you don't explain what you mean by "limit" here, and you cannot claim to have proved anything here if you don't explain what you mean by that. 

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decimal, representation, unique 
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