February 4th, 2016, 07:48 PM  #21 
Global Moderator Joined: Dec 2006 Posts: 20,617 Thanks: 2072  That's simply incorrect, as the phrase "all natural numbers" doesn't mean "from 1 to infinity". The article you linked to doesn't use the word "infinity". It refers to "all values of $n$". There are infinitely many possible values of $n$, but none of them is "infinity". 
February 4th, 2016, 07:48 PM  #22  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
https://en.wikipedia.org/wiki/Mathematical_induction For those who don't like or understand my proofs that the reals are countable, there is a "proof" that the reals are uncountable: https://en.wikipedia.org/wiki/Cantor...tability_proof Take your choice.  
February 4th, 2016, 07:53 PM  #23 
Global Moderator Joined: Dec 2006 Posts: 20,617 Thanks: 2072 
There is no such choice, as your assertions rely on a phrase that doesn't appear in the article you linked to. The linked article doesn't use the word "infinity".

February 4th, 2016, 07:53 PM  #24  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra  Quote:
Quote:
Everyone else here both knows and understands Cantor's proof. It's, again, just you that doesn't. Last edited by skipjack; February 4th, 2016 at 07:58 PM.  
February 4th, 2016, 09:35 PM  #25  
Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244  Quote:
$$\aleph_0\in\mathbb{N},$$ Or maybe just $$\infty\in\mathbb{N}.$$  
February 5th, 2016, 06:55 AM  #26 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Do you believe in the principle of mathematical induction?
Last edited by skipjack; February 5th, 2016 at 07:25 AM. 
February 5th, 2016, 07:19 AM  #27 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra 
The principal of mathematical induction is not like Father Christmas. It's a method that exists by logical deduction, not a story that you can choose to believe or not. If it were a story, you could make up whatever conclusion you wanted, but since it's not, you can't  which is something you don't seem to be able to come to terms with. Last edited by skipjack; March 5th, 2016 at 11:31 AM. 
February 5th, 2016, 07:26 AM  #28 
Global Moderator Joined: Dec 2006 Posts: 20,617 Thanks: 2072  You've already provided a link for mathematical induction, and the linked article doesn't use the word "infinity", so your proofs using induction and claiming to show that the reals are countable only show that terminating decimals are countable, which excludes all irrational reals.

February 5th, 2016, 08:15 AM  #29  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
.1234......... That's the 1:1 association. Whether or not you want to use the terminology "infinite" or "for all n" is academic.  
February 5th, 2016, 08:23 AM  #30  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra  Quote:
Perhaps you should investigate the difference between a set being countable ("listable" is a better term) and being able to say how many elements it has. Last edited by v8archie; February 5th, 2016 at 08:25 AM.  

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