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May 17th, 2018, 11:53 AM  #11 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 
The rationals are just another one dimensional vector field with an infinite number of vectors, which can be put into 1:1 correspondence with the reals, as I have shown (the reals are countable) By finite I mean having a max number. 1,2,3,4,........... is not finite 1,2,3 is. Oh oh, I better get out of here. 
May 17th, 2018, 12:38 PM  #12 
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  
May 18th, 2018, 08:26 AM  #13 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896 
zylo, did you really mean to say "the reals are countable"?

May 18th, 2018, 08:44 AM  #14 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124  
May 18th, 2018, 10:04 AM  #15 
Senior Member Joined: Sep 2015 From: USA Posts: 2,317 Thanks: 1230  
May 18th, 2018, 10:55 AM  #16  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 307 Thanks: 101 Math Focus: Number Theory, Algebraic Geometry  Quote:
Note that there's more to a vector space $V$ than just the set $V$; there's also the field of scalars $F$. For example, if $V = \mathbb{R}$ and $F = \mathbb{R}$, then $V$ is a 1dimensional vector space. However, if $V = \mathbb{R}$ and $F = \mathbb{Q}$, then $V$ is an infinite dimensional vector space. Your drivel about $\mathbb{R}$ being countable doesn't even dignify a response.  
May 19th, 2018, 03:58 AM  #17  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 896  Quote:  

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