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December 22nd, 2006, 10:14 AM   #1
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The Nature of Infinity

Quote:
Originally Posted by CRGreathouse
Quote:
Originally Posted by roadnottaken
Isn't the idea of a number greater that infinity nonsensical?
Yeah, pretty much. There are many ways to handle 'infinity', though:
  • The extended real line, probably what you think of when you say "infinity"[/*:m:19mdnr4i]
  • The projective line, where +infinity = -infinity[/*:m:19mdnr4i]
  • Hyperreal numbers, where there are numbers larger than all real numbers or strictly between all positive real numbers and 0[/*:m:19mdnr4i]
  • Surreal numbers, as described by Knuth (I'm still reading this one)[/*:m:19mdnr4i]
  • Cardinal numbers, where the number of integers = the number of integers + 1 = the number of rational numbers < the number of real numbers[/*:m:19mdnr4i]
  • Ordinal numbers, where the number of integers < the number of integers + 1[/*:m:19mdnr4i]
  • etc.[/*:m:19mdnr4i]

The interesting thing is that 'infinity' isn't a well-defined mathematical concept. Each area of mathematics has a different way of discussing it and dealing with it.

If you're interested in this, start a topic in the set theory/general area.
I find this an interesting topic. Does anyone have anything to add to it (including personal views on infinity)?
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December 22nd, 2006, 10:23 AM   #2
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So where should we start, O fearless leader?

I must admit, the lack of LaTeX is going to hurt a bit here. The set-theoretic infinities use non-Latin characters (omega, aleph, beth) not to mention different fonts (Fraktur? for c, the cardinality of the continuum).

Also, I don't see a way to attach files. Maybe if I feel the need for advanced formatting I'll just put a pdf up n my site for you.
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December 22nd, 2006, 10:40 AM   #3
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Yes, I can see how using a pdf file in your website might be easier than having to type out the names of greek and hebrew characters. Another possibility might be to create a topic in the AoPS forum, and post a link to it here. A drawback of that would be that people would have to register there in order to post, but it might make a good stop-gap (although I'm not sure if their $\LaTeX$ includes hebrew letters).
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December 22nd, 2006, 12:12 PM   #4
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I have set up this topic in the Art of Problem Solving forum.
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December 22nd, 2006, 12:23 PM   #5
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Sounds like a good stopgap for now. By the way, I'm known as Caradhras on AoPS forum. ( I sighed up there a while back, although I haven't done much there so far. )
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December 22nd, 2006, 12:30 PM   #6
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Man! It's nice to have Latex again...
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December 22nd, 2006, 01:54 PM   #7
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If you respond to this topic in the AoPS forum, but have a different username there than here, please post here with your username there.
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December 22nd, 2006, 02:33 PM   #8
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I don't have an account on that forum, but frankly making the whole thing in LaTeX (rather than just parts, as is usually done on the forums) is more attractive to me anyway.

http://math.crg4.com/infty.pdf
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December 22nd, 2006, 02:54 PM   #9
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If anyone has specific aspects of \infty that they are particularly interested in, I'm sure CRGreathouse will be happy to expand on those areas within the pdf, as well.
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December 22nd, 2006, 03:04 PM   #10
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What is the cardinality of the set consisting of the types of infinity?
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