September 17th, 2014, 08:26 AM  #11  
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  No. For example, neither $\omega$ nor $\omega^\omega$ are whole numbers. Quote:
 
September 17th, 2014, 08:27 AM  #12 
Member Joined: Aug 2014 From: Somewhere between order and chaos Posts: 44 Thanks: 5 Math Focus: Set theory, abstract algebra, analysis, computer science 
Well, after you get to infinity, you have an infinite progression of transfinite cardinals: $\displaystyle \aleph_0$, $\displaystyle \aleph_1$, $\displaystyle \aleph_2$, etc. Of course these are all still "infinity". Any number greater than an infinite number would have to be infinite itself, so the question of what lies beyond infinity is kind of meaningless.

September 17th, 2014, 08:33 AM  #13  
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  Quote:
The hyperreals (nonstandard analysis) and the ConwayKnuth surreals are probably the most common. The socalled "dual numbers" might be the easiest to understand. * Definitions differ on whether 0 is an infinitesimal. I think excluding it is most common.  
September 17th, 2014, 08:35 AM  #14 
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  Right. For any ordinal $\alpha$ there is a corresponding (much bigger!) cardinal $\aleph_\alpha$, so it's not just countably many alephs.

September 17th, 2014, 10:53 AM  #15  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Quote:
This suggests that there are three different zeros, +0, 0 and 0. Might the +0 and 0 be infinitesimals? Or how to understand this, perhaps there is an infinitesimal difference between 0 and 0 and between 0 and +0. I have arrived at this difference in this way: 10.999...... > 0 or 1  0.999........= +0 1 + 0.999.......< 0 or 1 + 0.999......= 0  
September 17th, 2014, 11:02 AM  #16  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra  Quote:
$0^+$ and $0^$ are notational conventions that show the direction of approach to (the unique) zero. They are not different numbers.  
September 17th, 2014, 11:18 AM  #17  
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  Quote:
Edit: Archie figured out why you were confused. $$ \lim_{x\to0^+} $$ just means that the path you take to get to 0 is from some point to its right. When you're taking limits in 2 or more dimensions you need a more complicated description of the path you take, but in 1 dimension all you need is a direction. Needless to say the path you take to get to a number is not the same as a number! Last edited by CRGreathouse; September 17th, 2014 at 11:51 AM.  
September 17th, 2014, 12:12 PM  #18  
Senior Member Joined: Nov 2013 Posts: 160 Thanks: 7  Quote:
they don't need to be the same numbers. The problem is to calculate what is this difference. My approach has been understanding the concept of limit process. I have managed to calculate the limits of single numbers. It may sound nonsense, but I have noticed it works. For example, the limit of 0 is not 0. So what is the limit of 0? The limit of 0 is the infinitesimal, and vice versa: the limit of the infinitesimal is 0. It is possible to choose a symbol for the infinitesimal so that it is easier to play with it, lets say it is x. Now we have: the limit of x is not x, the limit of x is 0. I have noticed that the rule "the limit of A is not equal to A" applies always, not matter what is the number A. Lately I found even very crazy developments: it is possible even to find the limits of the limits in this way. For example the limit of the limit of x is x. The limit of the limit of 0 is 0.  
September 17th, 2014, 03:45 PM  #19 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,654 Thanks: 2632 Math Focus: Mainly analysis and algebra  You have failed. Your attempt to create a difference between the three ways of writing zero was to use $\sum \frac{9}{10^n} = 1$, as you would know if you understood limits. But we went through all that a few of months ago, which is why I ignored it before. The rest of what you wrote is total nonsense. The limit of any constant $c$ as $x \to a$ is $c$. I'd actuallly be interested in finding out about respectable theories of infinitesimals. Last edited by v8archie; September 17th, 2014 at 03:57 PM. 
September 18th, 2014, 05:53 AM  #20  
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  Quote:
Quote:
Your conclusions about limits are incorrect. If you write out a formal proof you should be able to find your mistake. I suspect that you could not write a formal proof, though, so the process is not as easy as I make it sound.  

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