September 20th, 2018, 04:38 AM  #21 
Member Joined: Dec 2015 From: France Posts: 78 Thanks: 1  Last edited by skipjack; September 20th, 2018 at 06:59 PM. 
September 20th, 2018, 04:45 AM  #22 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,529 Thanks: 107 
So all natural numbers are finite?

September 20th, 2018, 04:50 AM  #23 
Member Joined: Dec 2015 From: France Posts: 78 Thanks: 1  Last edited by skipjack; September 20th, 2018 at 07:01 PM. 
September 20th, 2018, 05:09 AM  #24 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,529 Thanks: 107 
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, 06:42 AM  #25 
Senior Member Joined: Oct 2009 Posts: 544 Thanks: 174 
Damn, two Cantor disbelievers arguing with eachother. This is great stuff...

September 20th, 2018, 07:12 AM  #26 
Senior Member Joined: Dec 2015 From: Earth Posts: 245 Thanks: 27 
$\displaystyle y=1/n $ is simply more dense on $\displaystyle axis$ than $\displaystyle y=n$

September 20th, 2018, 02:06 PM  #27 
Member Joined: Dec 2015 From: France Posts: 78 Thanks: 1  
September 20th, 2018, 07:37 PM  #28 
Global Moderator Joined: Dec 2006 Posts: 19,700 Thanks: 1804  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, 06:11 AM  #29 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,529 Thanks: 107 
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, 06:29 AM  #30  
Senior Member Joined: Oct 2009 Posts: 544 Thanks: 174  Quote:
(Why am I bothering)  

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