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October 20th, 2016, 08:44 AM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,448 Thanks: 2123 Math Focus: Mainly analysis and algebra 
1) An infinite sequence of integers has an infinite number of elements. 2) For $\lim \limits_{n \to \infty} f(n)$, I prefer to say the limit as $n$ grows without bound. 3) It's false whatever the context is. 0.4999... = 0.5000... end of story. 4) I suggest you read your own post (#1) to see what you wrote. 5) No it isn't. It clearly has certain properties that your $P_n$ idea denies, such as having an infinite number of elements. If you wish to deny the existence of infinite sequences, you can no longer talk about Cantor's argument because there existence is part of the system in which he was working. Before you start claiming anything about your $P_n$ idea, you need to look at post #9 in particular. Last edited by v8archie; October 20th, 2016 at 09:10 AM. 
October 20th, 2016, 12:34 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 16,604 Thanks: 1201  You keep mentioning this, apparently with a view to justifying (eventually) your disagreement with it. However, you have never been able to state at precisely what stage Cantor makes any mistake, instead making somewhat vague claims along the lines of "he can't do that because...", but then writing about something that Cantor didn't do anyway. There is no point in defining something as a limit if you don't explain what you mean by "limit" in your particular context. Doing so makes your mathematical arguments imprecise and hence your conclusions are unproved. 
October 22nd, 2016, 07:50 AM  #13 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 915 Thanks: 72 
Consider the following list of onethousand nplace decimals: .3000000001 .3000000002 ............ .3000001000 Let the number of inbetween zeros approach infinity. Limit 1: Representation Limit. Each member of the list remains unique and becomes a real number. Limit 2: Analytic Limit . Each member of the list equals .3 Without distinguishing the limits, you have the impossible situation that the decimal representation of any number is not unique as n approaches infinity. Mathematicians fail to make the distinction which leads to unnecessary complication and confusion. An analogous situation is: Lim .4999999 = .499999........., Representation Limit Lim .4999999 = .5, Analytic Limit I addressed Cantor's argument a million times, in addition to all the other proofs of uncountability, in previous threads. In brief, any rational, intelligent discussion about properties at infinity has to begin with a discussion of finite properties at n, and if the conclusion holds for all n it holds for n > infinity. That is absolutely fundamental. Last edited by skipjack; October 22nd, 2016 at 10:11 AM. 
October 22nd, 2016, 11:18 AM  #14  
Math Team Joined: Dec 2013 From: Colombia Posts: 6,448 Thanks: 2123 Math Focus: Mainly analysis and algebra  How do you propose to do this? I don't see any way this can be achieved. The number of zeros is finite and must remain finite in order to have anything following them. It never gets any closer to being infinite and thus fails to "approach infinity". Quote:
This isn't true either, although every infinite sequences composed only of members of your lists converges to 0.3. That just shows that 0.3 is a real number. I don't think anyone is reeling in shock at that. Quote:
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But since you are trying to say that what holds for every finite case must hold for the infinite case, it is trivially false because the finite case is finite and the infinite case is infinite. So we immediately have a contradiction.  
October 22nd, 2016, 11:19 AM  #15  
Global Moderator Joined: Dec 2006 Posts: 16,604 Thanks: 1201  Quote:
Each member of what list? You are not defining either usage of "limit". Quote:
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If "for n > infinity" refers to a limit, you have failed to define that limit properly. Also, when a sequence has a properlydefined limit, it is typically the case that the limit has properties that aren't possessed by individual members of the sequence. For example, pi is the limit of various sequences of rationals, but pi isn't a rational.  

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