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December 4th, 2018, 05:01 AM  #1 
Newbie Joined: Dec 2018 From: Germany Posts: 5 Thanks: 0  Cardinality of subsets of naturals
Hi there, here's one question that's bugging me nuts. Cantor defined different kinds of infinity, named aleph_0, aleph_1, ..., each one the size of the powerset of a set of the previous size. The naturals, and therefore all infinite subsets of it, are countablyinfinite with cardinality aleph_0. Take now the set of primes. It's an infinite subset of the naturals. Now take the powerset of the primes, and it's not difficult to see a onetoone correspondence to the set of squarefree numbers: the unique factorization of each squarefree number corresponds to a subset of the primes. But the squarefree are also a subset of the naturals. So what gives. (Likely it's my understanding of Cantor's theory that is flaky.) Thanks! 
December 4th, 2018, 05:06 AM  #2  
Senior Member Joined: Oct 2009 Posts: 781 Thanks: 280  Quote:
The naturals, and therefore all infinite subsets of it, are countablyinfinite with cardinality aleph_0. Quote:
 
December 4th, 2018, 05:28 AM  #3 
Newbie Joined: Dec 2018 From: Germany Posts: 5 Thanks: 0 
Hi, thanks for the reply. (I think we have chatted before, back in the days were there was a number theory forum at PhysicsForums, if I'm not mistaken.) Ah, now I see your point  the squarefree correspond only to finite subsets of primes  all infinite subsets of primes are not accounted for. Now I can sleep, thanks 
December 4th, 2018, 06:00 AM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 609 Thanks: 378 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
 
December 4th, 2018, 06:28 AM  #5  
Senior Member Joined: May 2016 From: USA Posts: 1,310 Thanks: 551  Quote:
 
December 4th, 2018, 07:04 AM  #6 
Newbie Joined: Dec 2018 From: Germany Posts: 5 Thanks: 0 
So, just like the rest of the internet. Add a few work environments as well... Thanks all again 
December 4th, 2018, 09:05 AM  #7  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra  Quote:
Our intuitive view of counting is that it measures the number of elements in a set and thus the size. Cantor's genius was to see that this is not the truth. It works for our real world experience, just as Newton's laws of motion are fine for terrestrial applications of the pre20th century. But the truth is that a generalisation of the concept of size is more accurate on a larger scale, just as Einstein's generalisation of the laws of motion is more accurate. I often like to divorce the concept of cardinality entirely from that of "size" in my thinking and consider it as just some property. It makes understanding a little easier. It's not entirely accurate as cardinality is a measure of size, just not one that corresponds with our intuition when it comes to infinite sets.  

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