May 9th, 2017, 02:35 PM  #21  
Senior Member Joined: Jun 2014 From: USA Posts: 320 Thanks: 23  Quote:
Second, be considerate? Ahem... Zylo, et al., is obviously struggling. There is something more important here than just trying to get Zylo to admit that he/she doesn't have a valid disproof of Cantor's Theorem. IMHO, you need to stop ridiculing and back off. Until you do, you'll never get Zylo to listen to you (I wouldn't either).  
May 9th, 2017, 03:10 PM  #22 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,117 Thanks: 2369 Math Focus: Mainly analysis and algebra 
Zylo has been spouting that nonsense for the best part of a year. He has ignored every argument that doesn't fit what he wants to believe not just inthat thread, but in at least a dozen like it. He's not struggling to understand because he's not attempting to do so. He is just trying to muddy the waters for other students. I apologise for misunderstanding that a thread including $[0,1)$ in the title, every interval definition and your definition of A isn't supposed to include zero. Perhaps more clarity would help. Establishing that the cardinality of the set of infinite digit strings is the same as that of the reals seems to me to be much easier than trying to establish a bijection. 
May 9th, 2017, 03:29 PM  #23 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,117 Thanks: 2369 Math Focus: Mainly analysis and algebra 
Right on cue, he's started another one.

May 9th, 2017, 06:43 PM  #24  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,475 Thanks: 494 Math Focus: Yet to find out.  Quote:
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I really like his mention of assessing unformalised arguments, and i think the following sums everything up quite nicely... Quote:
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May 9th, 2017, 06:59 PM  #25  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,117 Thanks: 2369 Math Focus: Mainly analysis and algebra  Quote:
The representation $0.999\ldots$ means $$\sum_{k=1}^\infty \frac9{10^k}= \lim_{n \to \infty} \sum_{k=1}^n \frac9{10^k} = 1$$ or, equivalently, $0.999 \ldots$ is the limit of the sequence $\left( \frac9{10}, \frac{99}{100}, \frac{999}{1000}, \ldots, \frac{10^k1}{10^k} \ldots \right)$. Obviously we can do the same  finding the limit  with $1$. $1 = 1.000\ldots$ means $$1+ \sum_{k=1}^\infty \frac0{10^k}= \lim_{n \to \infty} \left(1 + \sum_{k=1}^n \frac0{10^k} \right) = 1$$ or, equivalently, $1.000 \ldots$ is the limit of the sequence $(1, 1, 1, \ldots)$. Limits can also be irrational. $$\pi = \sum_{k=0}^\infty (1)^k \frac4{2k+1} = \lim_{n \to \infty} \sum_{k=0}^n (1)^k \frac4{2k+1}$$ or $\pi$ is the limit of the sequence $(3, 3.1, 3.14, 3.141, \ldots)$. Indeed, one definition of Euler's number $e = 2.71828\ldots$ is $$e = \lim_{n \to \infty} \left(1 + \frac1n\right)^n$$ Last edited by v8archie; May 9th, 2017 at 07:16 PM.  
May 9th, 2017, 07:15 PM  #26  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,475 Thanks: 494 Math Focus: Yet to find out.  Quote:
 
May 9th, 2017, 07:35 PM  #27  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,117 Thanks: 2369 Math Focus: Mainly analysis and algebra  Quote:
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Last edited by v8archie; May 9th, 2017 at 07:37 PM.  
May 10th, 2017, 06:41 AM  #28  
Member Joined: Dec 2016 From: United States Posts: 53 Thanks: 3 Math Focus: Abstract Simulations 
So this is where I picked up the hint about space filling fractal curves. javascript  I need to make my function return a more organic collection of results  Stack Overflow Quote:
If the space can be filled continuously by the curve, then my novice intuition tells me the "potential space" of string combinations is also filled. The only thing left would be to find the right filling curve function. Last edited by InkSprite; May 10th, 2017 at 06:44 AM.  
May 10th, 2017, 07:12 AM  #29  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,475 Thanks: 494 Math Focus: Yet to find out.  Quote:
Most of the fractals in the list are 'pretty' looking fractals, a special sort of fractal. But in general, the key feature is 'selfsimilarity'. The idea that, regardless of scale, and object may exhibit the same structure. If you read the thread you posted, the first response lists various mathematical topics that are relevant here. Personally i think a good place to start, formally, would be a book by Steven Strogatz  "Nonlinear dynamics and chaos". From there you could progress to Edward Ott  "Chaos in dynamical systems".  
May 11th, 2017, 06:27 AM  #30 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,214 Thanks: 91 
If you remove the decimal from [ 0  1 ) what does that set match? The set of natural numbers. Proof: Assume there is a natural number not in the set. Put a decimal point in front of it. By definition, the numbers in [0,1) are the real numbers minus the integral part. The real numbers are countable. Ref: Cantors Diagonal Argument, Logic Real Numbers and Natural Numbers 

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