March 2nd, 2018, 07:23 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,462 Thanks: 106  Cantor's Infinite Binary Sequence
Let Ln be the list of nplace binary sequences: 00......00 nbinary digits 00......01 00......10 00......11 00....100 ............ 1111111 nones What is the difference between $\displaystyle \lim_{n \rightarrow \infty}$Ln and Cantor's list* of infinite binary sequences? * https://en.wikipedia.org/wiki/Cantor...gonal_argument Last edited by skipjack; March 3rd, 2018 at 01:37 PM. 
March 2nd, 2018, 08:28 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,403 Thanks: 2477 Math Focus: Mainly analysis and algebra 
In the limiting case, what do you think the second element in your list is?

March 2nd, 2018, 09:02 PM  #3  
Senior Member Joined: Sep 2016 From: USA Posts: 444 Thanks: 254 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
Last edited by skipjack; March 3rd, 2018 at 01:39 PM.  
March 3rd, 2018, 06:07 AM  #4  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,462 Thanks: 106  Quote:
I gave my definition of lim in OP. There are other definitions. So what? $\displaystyle \lim_{n \rightarrow \infty}$Ln is a list (representation) of the natural numbers in binary notation. If you add a period before each sequence you get a list (representation) of [0,1) in binary notation. How can a list of the natural numbers not be countable? Does anyone care to answer OP? EDIT On a line label a point 0 and another point 1. Then divide the line in half and label the points 0,1,2 respectiveley. Divide each half in half and then label the points 0,1,2,3,4. Keep repeating. In the limit you will have labelled each point on the segment with the natural numbers, the first after 0 always being 1. Last edited by zylo; March 3rd, 2018 at 06:27 AM.  
March 3rd, 2018, 07:28 AM  #5  
Senior Member Joined: Sep 2016 From: USA Posts: 444 Thanks: 254 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
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Last edited by skipjack; March 3rd, 2018 at 01:42 PM.  
March 3rd, 2018, 08:34 AM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,403 Thanks: 2477 Math Focus: Mainly analysis and algebra  Quote:
There is no first point after the endpoint of a line. By definition, if you claim that you have that first point, I can always show you one that comes before it. Again, there is no limit with every point labelled. If you think you have labelled them all, you could give me two adjacent points, and I will show you a point between them. My question was not about the "value" of your second element. I want to know what you think it consists of. It's clearly some zeros followed by a 1. How many zeros? It can't be an infinite number of zeros because that means that the zeros never end and thus can have nothing following them. So my position is that you are unable to construct your limiting list and therefore it is a fallacy to assume that it exists. You first have to show that the list exists before you can draw any conclusions. If you do manage to show that the limiting list is well defined, I'll give you the conclusion that you draw from the assumption that is does. Good luck with that, it's already been proved not to exist. So you are going to need a different mathematical system. Last edited by v8archie; March 3rd, 2018 at 08:39 AM.  
March 3rd, 2018, 11:10 AM  #7 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,462 Thanks: 106 
0 0 0 1 numerical value 2^0 0 0 1 0 numerical value 2^1 0 1 0 0 numerical value 2^2 1 0 0 0 numerical value 2^3, no matter, in general, how many zeros precede the first 1. Google "binary number". Since no one seems able or willing to address or answer OP, Hint: Infinite binary sequence: $\displaystyle \lim_{n \rightarrow \infty} a_{1}a_{2}......a_{n}$, $\displaystyle a_{i}$ = 0 or 1 Last edited by skipjack; March 3rd, 2018 at 01:44 PM. 
March 3rd, 2018, 11:42 AM  #8 
Senior Member Joined: Jun 2014 From: USA Posts: 366 Thanks: 26  Someday you'll look back on this and realize how patient and willing everyone has been with you... well, maybe. I hope so anyways. I can personally relate to being both wrong and stubborn for what that's worth too. It's when you cross over into being delusional and obsessed that it's cause for concern. I personally don't have much time to spend leisurely here at mymathforum so I'm not here often, but I decided to browse last night and there you were making the same old arguments yet again. I will refrain from responding to you when it comes to 'Cantor stuff' from now on simply because I think attempting to do so is doing you a disservice. I hope you answer the questions that I posed for you in your other thread and I hope others help you in that regard, but beyond that, I strongly suggest others refrain from entertaining your attempts to assert what is essentially the same thing over and over and over again. 
March 3rd, 2018, 11:52 AM  #9 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,462 Thanks: 106 
Yet another offtopic reply. I answered the first two offtopic questions out of courtesy, the others should be obvious from what I have said. If not, start your own thread. Anyone with an answer to OP? 
March 3rd, 2018, 01:29 PM  #10  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,403 Thanks: 2477 Math Focus: Mainly analysis and algebra  Quote:
I've done nothing but address you original post. But you've now, once again, resorted to ignoring points made and questions posed in favour of repeating nonsense. Last edited by v8archie; March 3rd, 2018 at 01:52 PM.  

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