July 13th, 2017, 03:28 PM  #11 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,032 Thanks: 2342 Math Focus: Mainly analysis and algebra  
July 13th, 2017, 04:09 PM  #12 
Senior Member Joined: Oct 2009 Posts: 142 Thanks: 60  Sure, the sequence does not converge in the usual meaning of convergence. But it can sometimes be very very useful to change the usual meaning of convergence to get good limits. Yes, it's a nasty piece of mathematical trickery, but it's very common and useful. This is what the 2adics do.

July 13th, 2017, 05:38 PM  #13 
Senior Member Joined: Aug 2012 Posts: 1,636 Thanks: 413  
July 13th, 2017, 07:17 PM  #14 
Senior Member Joined: May 2016 From: USA Posts: 824 Thanks: 335 
If I remember a long conversation with maschke about a year ago (on a zylo thread of course), it is sufficient to show that the possible number of infinite bit strings exceeds the number of natural numbers using the diagonal proof without any reference to the representation of real numbers. In other words, if infinity "exists," it "exists" in different flavors that are not commensurate, where commensurate means finding a means to put two sets into onetoone correspondence. (That proof alone does not show that the set of real numbers exceeds the set of number of natural numbers, but it prevents a lot of confusion about the diagonal proof itself. Dealing with the number of real numbers then requires a separate proof.) It is perfectly reasonable to declare that there is no physical evidence for any kind of infinity nor any physical evidence of irrational numbers. Finitists have a respectable empirical basis for their position. Maschke and I disagreed (I can't remember why) about my premise that infinity certainly does exist as a mental construct, a creation of the human imagination. But if you insist on a demonstration that infinity is physically meaningful, then the natural numbers cannot be shown to be infinite. You can't have it both ways: either we agree to accept that infinity "exists" or, by denying that infinity "exists," to concede that the natural numbers are not infinite. 
July 13th, 2017, 07:44 PM  #15 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,032 Thanks: 2342 Math Focus: Mainly analysis and algebra  
July 14th, 2017, 02:48 AM  #16  
Global Moderator Joined: Dec 2006 Posts: 18,154 Thanks: 1421  Quote:
If there are an infinite number of nonzero terms, how do you define their sum, zylo, given that every natural number (such as 4 in the example) already corresponds to such a sum with a finite number of nonzero terms?  
July 14th, 2017, 03:58 AM  #17 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,410 Thanks: 481 Math Focus: Yet to find out.  
July 14th, 2017, 09:17 AM  #18 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,175 Thanks: 90  https://en.wikipedia.org/wiki/Padic_number padics: Abstract irrelevance. Std Trick 1. Subtraction is not defined for the natural numbers. I never claimed removing binary point from an infinite binary fraction was a mathematical operation. I simply said if an infinite binary sequence exists after a binary point, it exists without the binary point. Same trick over and over again: Joe saya the moon is made of green cheese implying Joe is a jerk. But Joe never said the moon was made of green cheese, proving what about the person who made the false attribution? Std Trick 2. Personally, I consider Tricks 1 and 2 as failure of the propounders to find anything wrong with OP. $\displaystyle All$ binary fractions are infinite, as are all decimal fractions. .840 is ambiguous. It is either exactly .840, ie, .840000000........(an infinite decimal fraction) or a number between .839500.... and .840500......., (infinite decimal fractions). EDIT: What defines a binary fraction: A unique infinite sequence of binary digits. The same unique infinite sequence of binary digits defines a natural number. Last edited by zylo; July 14th, 2017 at 09:31 AM. 
July 14th, 2017, 09:42 AM  #19  
Senior Member Joined: Oct 2009 Posts: 142 Thanks: 60  Quote:
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You're not talking about decimal points. But yes, I agree, .840 is simply shorthand for .84000000... But it is a useful and often used shorthand.  
July 14th, 2017, 10:53 AM  #20 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,175 Thanks: 90 
The infinite binary (or decimal) sequence $\displaystyle a_{1}a_{2}a_{3}.......$ defines a unique natural number or unique binary fraction, depending on interpretation: Natural Number: $\displaystyle \sum a_{n}2^{n}$ which approaches a unique natural number as n > infnity. Binary Fraction: $\displaystyle \sum a_{n}2^{n}$ which approaches a unique fraction in [0,1) as n > infinity. Note the unique association between natural number and binary fraction. 

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