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Define negative numbers to be greater than infinity 
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#1
Jul809, 07:38 PM

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#2
Jul809, 07:53 PM

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Just what does it mean to be "greater than infinity"? 


#3
Jul809, 10:49 PM

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#4
Jul809, 11:03 PM

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Define negative numbers to be greater than infinity
This sounds like the end result one would obtain by applying twoscompliment arithmetic to the set of {reals, infinity}.



#5
Jul809, 11:18 PM

P: 48

Let a be any negative real number, and b be any positive real number.
a < b Hence, how in the world is negative numbers greater than infinity?! 


#6
Jul809, 11:49 PM

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Because, in this paper, they define a new order relation (I'll write <<) such that 0 << 1 << 2 << 3 << ... << 3 << 2 << 1. In this way, for any negative a and positive b, we have b << a. If we were to add an element infinity to this, then we would have b << infinity << a for any negative a, positive b.



#7
Jul909, 12:16 AM

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#8
Jul909, 03:13 AM

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It sounds like complete nonsense.
To begin with, the authors seems to confuse concepts like axioms and conditions. 


#9
Jul909, 08:13 AM

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#10
Jul909, 09:17 AM

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They're giving up transitivity, which is a pretty big blow. What does their system gain?



#11
Jul909, 09:32 AM

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I only had a quick look yesterday. It seems to me that the gain is that you have a more efficient formalism for doing computations involving divergent series. 


#12
Jul909, 12:36 PM

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I'm stuck on definition 2.1. How is that supposed to work for 0?



#13
Jul909, 12:50 PM

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#14
Jul909, 01:14 PM

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I looked it over. There is some interesting material in there. The "new ordering" isn't the significant part.
I think it works to their disadvantage to use existing notation like [tex]\sum_{n=a}^b f(n)[/tex] with their new meaning. Better would be using a different notation. 


#15
Jul909, 01:48 PM

P: 33

it doesnt make sense simply because it is a different mathematical system than the one weve become accustomed to, you cant compare its results with traditional mathematical problems because the value of infinity is more "numerous" than a negative. its abstract in a way that makes less realistic sense but more ordering efficiency. just as imaginary numbers are used in situations when real numbers cannot provide a solution.



#16
Jul909, 04:48 PM

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I do think you guys are being too hard on them. Constructing linear operators that extend the domain of summation is not that uncommon. I doubt the ordering on Z that they use is actually relevant  it just for whatever reason happened to suggest a path.



#17
Jul1209, 08:01 AM

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#18
Jul1209, 08:09 AM

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If they had said that they had been INSPIRED by that view to construct a new number system, rather than pushing for its "correctness", I would have been less suspicious of it. I haven't bothered to look much further into it, I'll admit. 


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