July 6th, 2018, 03:35 PM  #51 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,721 Thanks: 599 Math Focus: Yet to find out. 
Why decimal representation exactly? The heart of your argument seems to claim that we can't, with any level of confidence, assert things about processes involving infinity. Do you take issue with other infinite sums that have a convergent value? Anyway, why doesn't someone start a Google doc to write out all the arguments for and against in concise form with in depth explanations? There's loads of repeated information here that could be made better use of. 
July 6th, 2018, 03:49 PM  #52  
Senior Member Joined: Aug 2012 Posts: 2,045 Thanks: 584  Quote:
Quote:
Quote:
https://en.wikipedia.org/wiki/Real_number  
July 6th, 2018, 07:05 PM  #53  
Senior Member Joined: Aug 2012 Posts: 2,045 Thanks: 584  Quote:
The real numbers manifestly fail this standard. We can write $\displaystyle \mathbb R = \bigcup_{x \in \mathbb R} \{x\}$. If we can break up the real numbers into a union of singletons, none of which are anything like the full set of real numbers ... then the mathematical real numbers are not the right model of the continuum. I may be paraphrasing Peirce badly here, my understanding is second or third hand. I find this a compelling line of argument regardless. The entire Cantorian and settheoretic paradigm cannot lead us to the continuum. Perhaps this is something to to with the rise of Category theory and Homotopy type theory as potential alternative foundations. Peirce as I understand it was a pretty deep thinker, not as famous as some, but who perhaps anticipated the modern developments in foundations. https://en.wikipedia.org/wiki/Charles_Sanders_Peirce Last edited by Maschke; July 6th, 2018 at 07:14 PM.  
July 6th, 2018, 10:31 PM  #54 
Global Moderator Joined: Dec 2006 Posts: 19,714 Thanks: 1806  Is a nonrepeating decimal a type of decimal? If not, how do you define it? How can you count them? You are assuming that each ".33...." isn't endless. That allows 1/3 to be represented by the endless .3333333........, doesn't it? How would counting be done without a number system? (I asked this before, but you didn't tell me.) 
July 7th, 2018, 05:26 AM  #55 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,546 Thanks: 110 
The axiomatic definition of a real number doesn't let you calculate $\displaystyle \oint_{0}^{\frac{1}{3}}f(x)dx$ What bothers me is that neither does the decimal number system because 1/3 doesn't exist there, and though [0,1/3) exists, [0,1/3] doesn't. You can waffle around it, but you really don't have an unambiquous number system at your disposal. The rational numbers plus infinite decimals seem to cover all points on a line to the extent that they define it in an unequivocal manner, but you are really dealing with two number systems with duplications. But maybe two number systems combined (rational and infinite decimal) isn't so bad, since it means you can always specify a point on a line without equivocation. And with that, 3(1/3)=1 $\displaystyle \neq$ .999....... And, brace yourself, since rational and reals are countable, so is their combination, even with duplication. Induction is infinity. As for something like pi, whose only numerical definition is it's decimal expansion, there is no ambiguity. One, two, three, four, ........, ie, with the natural numbers Last edited by zylo; July 7th, 2018 at 05:31 AM. 
July 7th, 2018, 08:31 AM  #56  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,445 Thanks: 2499 Math Focus: Mainly analysis and algebra  Quote:
You don't want to use the definitions and you are claiming erroneously that decimal strings are numbers. Decimal strings can be used to represent numbers. There's a huge difference. Moreover, you refuse to learn anything about infinite objects. You insist on relying on your intuition about the infinite and your intuition is wrong. As is everybody's. As a result nothing you say about infinite objects is correct.  
July 7th, 2018, 08:53 AM  #57 
Global Moderator Joined: Dec 2006 Posts: 19,714 Thanks: 1806  Isn't "the natural numbers" a number system? Please answer my question. How would counting be done without a number system? If you have a method, please describe how it would be used to count 1, 2, 3, 4, ... (continued without end). 
July 7th, 2018, 10:56 AM  #58  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,546 Thanks: 110  Quote:
Edit: Trying to guess what you are getting at 0 one 1 two 00 one 01 two 10 three 11 four 000 one 001 two 010 three 011 four 100 five 101 six 110 seven 111 eight . . 9999999........ They are defined for all n, ie, they are defined for n+1 when they are defined for n, ie, in the limit as n > infinity, when they represent infinite divisions of a line or, with a period in front of them the real numbers (without some rationals like 1/3)* in [0,1), or as another notation for the natural numbers (all of them as n > infinity). In any event, countable. * As explained in post #55 Last edited by zylo; July 7th, 2018 at 11:28 AM.  
July 7th, 2018, 12:01 PM  #59 
Global Moderator Joined: Dec 2006 Posts: 19,714 Thanks: 1806 
Perhaps you hadn't noticed the word "without". My question remains: how would counting be done without a number system? I'm not asking how you would use a number system to do counting. I didn't quite understand your example anyway... which column showed what you were counting, given that neither column seems to contain 1, 2, 3, 4, ... (continued without end)?

July 7th, 2018, 12:19 PM  #60 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,546 Thanks: 110 
Sticks, stones, scratches on a rock, etc I was counting nplace binary representations, countable for all n through n approaches infinity, as proved by induction. 

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