February 4th, 2019, 05:46 AM  #81  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
2) Assumption: The mirror image of the digits in any real number in (0,1) exists. 3) Finite: ends. Infinite: doesn't end, endless. How is it, that by taking one step at a time, I can embark on an endless journey?  
February 4th, 2019, 07:06 AM  #82  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 312 Thanks: 111 Math Focus: Number Theory, Algebraic Geometry  Quote:
[(*) If it ended, it would have an element of maximal length, say the string consisting of n 1s, where n is some (finite) natural number. But then the string consisting of n+1 1s would be a longer string, and is still in the list as n+1 is also a (finite) natural number. This is a contradiction.] What this example shows: the endlessness of a list of strings does not imply that the list contains strings of infinite length.  
February 4th, 2019, 08:48 AM  #83 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,659 Thanks: 2635 Math Focus: Mainly analysis and algebra  
February 4th, 2019, 09:33 AM  #84  
Global Moderator Joined: Dec 2006 Posts: 20,636 Thanks: 2081  Quote:
Quote:
(3) From your starting point of 1, your steps each involve adding 1, which is welldefined only for adding 1 to a finitelength number, and always gives a finitelength number. You can't reach 545454545... (which isn't finitelength) by adding 1, so you don't produce "all sequences of digits".  
February 4th, 2019, 10:05 AM  #85 
Senior Member Joined: Dec 2015 From: somewhere Posts: 514 Thanks: 80 
This is simply not defined yet. Which one is larger 222222... or 999999... (the two numbers are natural), but $\displaystyle \frac{2222...}{9999...}=2/9=0,\overline{ 2}$. Last edited by skipjack; February 4th, 2019 at 10:35 AM. 
February 4th, 2019, 10:43 AM  #86 
Global Moderator Joined: Dec 2006 Posts: 20,636 Thanks: 2081 
The second number has greater magnitude, but the lengths of the decimal representations are equal.

February 4th, 2019, 10:46 AM  #87  
Math Team Joined: May 2013 From: The Astral plane Posts: 2,160 Thanks: 878 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
aka The fraction does not exist except in zylo's system of numbers. Dan  
February 4th, 2019, 10:49 AM  #88 
Global Moderator Joined: Dec 2006 Posts: 20,636 Thanks: 2081  
February 4th, 2019, 12:55 PM  #89 
Math Team Joined: May 2013 From: The Astral plane Posts: 2,160 Thanks: 878 Math Focus: Wibbly wobbly timeywimey stuff.  

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