July 20th, 2009, 07:16 AM  #1  
Member Joined: Jun 2009 Posts: 62 Thanks: 0  infinity Quote:
 
July 20th, 2009, 07:47 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: infinity
Set theory has a wide variety of infinite numbers. These come in two types: cardinals and ordinals. Cardinals can be thought of as counting the number of elements of a set. For example, denote by S the cardinality of the set S. Then {1, 7} = 2. If Z is the set of integers and R the set of real numbers, then Z = aleph_0 and R = 2^aleph_0 > aleph_0. (This inequality is called Cantor's theorem.) There are also the extended real numbers (similar to the infinity symbol in calculus, comes in two flavors: +infinity and infinity), the projective real numbers (only one type of infinity, which can be thought of as 'the smallest projective real' as well as 'the largest projective real'). There are also the hyperreals, the surreals, and many others. 
July 21st, 2009, 02:46 AM  #3 
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: infinity
Two sets are defined as being the same size if it's possible to exhibit a bijection (one to one correspondence) between their elements. Also, for a set define . It's possible to show that is strictly greater than (an injection is possible but no bijection) for all sets . So, if we start with an infinite set, such as the set of all natural numbers, we must end up with a bigger set. This bigger set gives us a bigger (cardinal) infinity. We can then do the same thing to the new set to obtain another infinity and so on. This is not the whole story by any means but it does provide a whole pile of different infinites. There are others too but they are more difficult to explain.

July 21st, 2009, 06:20 AM  #4  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: infinity Quote:
 
July 21st, 2009, 07:49 AM  #5  
Senior Member Joined: Nov 2008 Posts: 199 Thanks: 0  Re: infinity Quote:
 
July 22nd, 2009, 07:29 PM  #6  
Member Joined: Jun 2009 Posts: 62 Thanks: 0  Re: infinity Quote:
 
July 22nd, 2009, 07:32 PM  #7  
Member Joined: Jun 2009 Posts: 62 Thanks: 0  Re: infinity Quote:
 
July 22nd, 2009, 07:42 PM  #8  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: infinity Quote:
Basically, let's say you understand the smallest infinite ordinal ? = {0, 1, 2, 3, ...}. You're allowed to write ? + 1, the 'next' ordinal after ?. Keep going in this way and you get ? + 2, ? + 3, ..., ? + ? = ? * 2. After that comes ? * 2 + 1, ? * 2 + 2, ..., ? * 3. You can keep going in like fashion: ? * 3, ? * 4, ..., ? * ? = ?^2. Like that? OK, it's pretty clear that you can write a bunch of ordinals in like fashion: ?^k * a_k + ... + ?^3 * a_3 + ?^2 * a_2 + ? * a_1 + a_0. This is (a subset of) Cantor normal form. It's easy enough to compare two numbers in this form: just find the highest power of ? and take the one with the greater coefficient. If they're the same, move on to the nexthighest and so on. Of course if we're going to write ?, ?^2, and so forth you just know that ?^? is coming. And from there it's a small step to ?^(?^?). Using that ellipsis that you know and love, I'll write ?, ?^?, ?^(?^?), ..., ??.  
July 26th, 2009, 07:43 PM  #9  
Member Joined: Jun 2009 Posts: 62 Thanks: 0  Re: infinity Quote:
 
July 26th, 2009, 08:07 PM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: infinity Quote:
Quote:
 

Tags 
infinity 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
is 1/infinity = lim x>infinity 1/x  sivela  Calculus  1  June 25th, 2012 10:04 AM 
Infinity (?)  kfarnan  Calculus  3  August 29th, 2011 10:47 AM 
Infinity...  nerd9  Algebra  22  July 18th, 2010 03:47 PM 
Solve X where infinity>x>infinity  DQ  Algebra  5  September 14th, 2009 05:13 AM 