Infinity

[stanek]

What is infinity? Something that goes on forever.

Can you prove something is infinite? I believe in infinity but I do not know how to prove it.

How do you imagine infinity? I imagine a number and add another number and imagine doing that forever.

Is pi infinite? If I imagine pi as the curve of a circle I get a tight spiral that never quite connects. I think this comes from the more precise pi is the tighter the circle becomes.

Does anyone have any recommended readings or web pages on infinity?

I look forward to hearing your answers, as well as better questions too. My answers are in italics.

[Siegmund]

I'll take a stab at this, even though I know I'll be sorry in the morning.

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What is infinity? Can you prove something is infinite?

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It's most easily defined negatively: that which is not finite. "Finite" usually gets defined something like "there exists a fixed number N, such that every number with such-and-such property is less than N / such that the object fits inside the region [0,N]x[0,N] / (or whatever the exact description of your finite item is)". That proof often entails showing that some assertion or another is false for all k>N.

You can, if you want, turn that inside-out, and use some sort of direct "if there is some N such that all k>N have this property, then the set of numbers having this property is infinite" - but people tend to have trouble correctly pinning down the details of "for all" quantifications.

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How do you imagine infinity?

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To be honest, I rarely have the need. It's a lot harder for me to form a concrete image in my head of a billion, or of 1326, or even 25, that it is for me for say "sufficiently big that I don't care exactly how big it is" -- by which I mean, I have no need of a fixed reference point N like I do for a large finite set. For almost all purposes, a simple set of concepts like "zero, one, a few, so many they all blur together" are all that are needed.

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Is pi infinite?

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In the sense that both you and I answered the first two questions, no; it's very clearly more than 3.1 and less than 3.2, and I can quickly and simply construct it by wrapping a string around a unit circle.

But you DO bring up a different KIND of 'infiniteness': the kind above might be better termed 'unboundedness', and this kind might be termed 'fuzziness even under arbitrarily high magnification', or more simply 'irrationality.'

Your image, or any of several similar images (asymptotes, successive zoom-ins on fractals, etc) works well for visualizing this kind of non-simple behaviour.

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Does anyone have any recommended readings or web pages on infinity?

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One reasonable presentation is in a moderately early chapter of Hofstadter's Godel Escher Bach - approximately the section where he talks about Bloop Floop and Gloop - also a good place to get used to the "there exists" and "for all" quantifiers.

There are lots of bad places ... in your particular case, since you're looking for something that provides you with something different than "whatever it is, add one to it," you will probably NOT gain much from yet another look at the old saws like Euclid's infinitude of primes or Hilbert's Hotel.

I would suggest a key idea worth concentrating on, which many people, even very good math students, have trouble with, is grasping the difference between "if you give me an example of one of your sets, I will find an N that bounds it" (not a proof of finiteness of a family of sets) and "I will give you an N, and let you pick one of your sets to see if it fits" (a valid proof of finiteness).

[Philo]

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Does anyone have any recommended readings or web pages on infinity?



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http://www.hup.harvard.edu/catalog/LAVUND.html

[LuckOfTheDraw]

When I was a child, I had a dream where a voice told me to, "Imagine infinity, infinity, infinity..." and kept repeating "infinity". I felt as if my brain was stuck trying to imagine it, although my mind was absolutely blank, as if hypnotized by the repeating word. I had a distinct feeling that I was going to die if I couldn't snap out of it. I woke up with a jolt. Looking back on to the experience, it was very similar to later episodes I had with sleep paralysis and night terrors. Very strange memory.