September 20th, 2018, 05:38 AM  #21 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Last edited by skipjack; September 20th, 2018 at 07:59 PM. 
September 20th, 2018, 05:45 AM  #22 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
So all natural numbers are finite?

September 20th, 2018, 05:50 AM  #23 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Last edited by skipjack; September 20th, 2018 at 08:01 PM. 
September 20th, 2018, 06:09 AM  #24 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
Clearly we have to define finite. Finite sequence: one that ends I note that the sequence of natural numbers (0,1,2,...) does not end. 
September 20th, 2018, 07:42 AM  #25 
Senior Member Joined: Oct 2009 Posts: 629 Thanks: 192 
Damn, two Cantor disbelievers arguing with eachother. This is great stuff...

September 20th, 2018, 08:12 AM  #26 
Senior Member Joined: Dec 2015 From: Earth Posts: 274 Thanks: 31 
$\displaystyle y=1/n $ is simply more dense on $\displaystyle axis$ than $\displaystyle y=n$

September 20th, 2018, 03:06 PM  #27 
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  
September 20th, 2018, 08:37 PM  #28 
Global Moderator Joined: Dec 2006 Posts: 19,977 Thanks: 1851  It suffices to understand "enumeration", "infinitely many" and "infinitely long", as what's being referred to is that any enumeration of infinitely many infinitely long binary sequences allows an infinitely long binary sequence to be specified (by "diagonalization") that isn't in the enumeration.

September 22nd, 2018, 07:11 AM  #29 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
It's easy to set up a 11 correspondence between infinite binary digits and the natural numbers 0: 00000....... 1: 10000...... 2: 01000...... 3: 11000...... 4: 00100....... 5: 10100....... 6: 01100...... . . 
September 22nd, 2018, 07:29 AM  #30  
Senior Member Joined: Oct 2009 Posts: 629 Thanks: 192  Quote:
(Why am I bothering)  

Tags 
binaryexpressed, cardinality, continuum hypothesis, diagonal argument, numbers, real, set 
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