April 10th, 2016, 11:50 AM  #1 
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2  Contradiction about the Set of Natural Numbers? It is said that there is an infinite number (aleph 0) of elements in the set of natural numbers. If we are counting the natural numbers in such a way as to have a matching of a onetoone correlation where n = x; for example, if the 5th element in the set of natural numbers is n = 5, then x = 5. But, if we have an infinite number of elements, where x = aleph 0, then n = aleph 0, but n must be a natural number. Last edited by Mathbound; April 10th, 2016 at 12:09 PM. 
April 10th, 2016, 12:03 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,620 Thanks: 2608 Math Focus: Mainly analysis and algebra 
You are confusing ordinals and cardinals. You are also ignoring the definition of the natural numbers. In particular, there is no axiom or definition that says that either cardinal numbers or ordinal numbers must be natural numbers.

April 10th, 2016, 12:20 PM  #3 
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2 
But n must be a natural number: N = {1x=1, 2x=2, 3x=3, ...} I should have been clearer. 
April 10th, 2016, 12:31 PM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,620 Thanks: 2608 Math Focus: Mainly analysis and algebra 
Why must it? I can match the natural numbers to the natural numbers easily enough, but there are no infinite natural numbers, so no natural number describes the size of the set of natural numbers because the set is infinite.

April 10th, 2016, 02:42 PM  #5 
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2  That's the problem; there can't be a match of aleph 0 number of elements x to anything in the set. But the set is said to have an aleph 0 number of elements.

April 10th, 2016, 02:58 PM  #6 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,620 Thanks: 2608 Math Focus: Mainly analysis and algebra 
Yes. What's the problem? $\{0,1,2,3\}$ has four members, but 4 is not a member of the set. And some people define the natural numbers to include zero (indeed, it's the most natural settheoretic definition). To have a natural number represent the size of the set of natural numbers, you'd need a natural number that is bigger than all the rest. And that isn't possible by the definition of natural numbers. 
April 10th, 2016, 06:28 PM  #7  
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2  Quote:
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April 10th, 2016, 06:38 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,620 Thanks: 2608 Math Focus: Mainly analysis and algebra 
All you are doing is trying to change the definitions so that they fit your assertion. In particular, if $x$ doesn't have to be a natural number, why should $n$ have to be a natural number? Not that is makes any difference. The set $\{\aleph_0, 1, 2, 3, ...\}$ is also infinite and has cardinality $\aleph_0$. I think you might be trying to play with ordinals, in which case you should use $\omega$ rather than $\aleph_0$. 
April 10th, 2016, 07:29 PM  #9  
Member Joined: Jun 2014 From: Alberta Posts: 56 Thanks: 2  Quote:
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April 10th, 2016, 08:13 PM  #10  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,620 Thanks: 2608 Math Focus: Mainly analysis and algebra  Quote:
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No, it's a cardinal number which means it describes the size of a set. An ordinal describes the ordering of elements of a set. Perhaps this video will help. (With thanks to whoever posted it this morning). Last edited by skipjack; April 11th, 2016 at 11:24 PM.  

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