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Comment by yannis | Hacker News Reader

One side of my brain accepts and understands the Maths but another corner of it can never understand the paradox as to how one infinity is greater than the other; since the reals are uncountable that is, there are strictly more real numbers than natural numbers, and both sets are infinite.

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Interesting - thanks. I wonder if the problem is that people try to think of infinity as a number that's just really, really big - bigger than all the numbers. If you think of infinity as being the biggest thing there is, then you've got a problem.

You (and others) are not alone. Infinity was a major problem for a very, very long time, with many contradictions and paradoxes arising because of insufficiently careful reasoning.

Here's another explanation:

http://www.solipsys.co.uk/new/CantorVisitsHilbertsHotel.html...

and here's something some people think of as a paradox:

http://www.solipsys.co.uk/new/BallsInBarrels.html?HN

I made an interesting observation recently, and I think it's not reinforced enough in the teaching of maths: "infinite" and "infinity" have less to do with each other than you would think. "Infinity" is a particular point in a set. "Infinite" describes the size of a set. There's only one "infinity"[1] but there's (infinitely!) many types of "infinite." By application of the Sapir-Whorf hypothesis, this would be easier for people to understand if we used more separate terms for the two concepts.

[1] Well, you can have +/-infinity... or you can make it the same infinity at both ends. This is really the only way to go when you deal with the complex plane instead of the real line. But regardless, the point is in topology, infinity can be treated as singular and definite.

You have an interesting and perhaps useful conceptual point, but your language is at odds with current usage.

The concepts of infinite sizes of sets and point "at infinity" are different, and should be separated.

The thing is, there are multiple infinities even in the sense you're thinking about. For example, you can have one at the end of every line in the complex plane. That gives you the projective plane. Again, in topology.

The one you're thinking of is the one-point compactification of the plane, but it's not the only one.

Actually, having an infinity at the end of every line on the complex plane just gives you a simple disc. To get the projective plane, you have to associate antipodal points, like associating +/-infinity on the real line to a single infinity, but for every line that runs through the origin separately. But in topology, whether these points are 'at infinity' or not is arbitrary, because the complex plane is indistinguishable from the unit disc, until you start adding some more structure to it like a differential structure.

And trust me, you do not want to be doing any sort of calculus on the complex plane with anything other than a single infinity. Conformal mappings really don't work well without that simplification. And with that simplification, everything is magical and you can integrate rainbows to get unicorns and candy canes.

Nonetheless, your point is valid, it's possible to have more than a single point at infinity. However, regardless, it's more conventional to think of infinity as a particular point with some unusual properties, and in these cases with multiple infinities they're all basically the same. You don't have classes of infinities the same way you have classes of infiniteness.

  > Actually, having an infinity at the end of
  > every line on the complex plane just gives
  > you a simple disc.

Er, judging from the rest of your comment you must simply have misunderstood me. For every line through 0 in the complex plane, take a point and define it to be at both ends of the line. That is doing as you say, associating antipodal points.

And I do do calculus (well, equivalent procedures) on such objects. It's clear from your comments that you know about these things - trust me that I do as well.

So we're agreed that the concept of infinite sized sets and the concept of "points at infinity" are different, and for teaching about "infinity" it might be useful to separate them explicitly.

Interesting observation. Thanks.

Glad we cleared that up, and glad I could provide some insight =).

Also, thanks for being civil in response to my show-off-y reply.

It is also a matter of being clear about some terminology early during discussions, if one says for example that the set of integers is countably infinite and that the set of real numbers is uncountably infinite is easier to understand.

The Aleph Numbers are the terms to describe the types of infinity to which you are referring:

http://en.wikipedia.org/wiki/Aleph_number

I'd like to chime in here as someone who is math-interested, but never went beyond high school maths (analytical trig, sadly).

IIRC, I first got my head around different orders of magnitude regarding infinities with George Gamow's popular "1, 2, 3, infinity" [1].

However -- I think, it's been a long long time since I read it -- that book presented different infinities in exactly the same way that someone who could only count to 20 would compare sets of hundreds of items.

That is: assume that space is infinitely bounded, with an infinite amount of matter. You might say that there are infinitely many galaxies, and there are infinitely many stars. However, there are also infinitely many more stars than there are galaxies, because each galaxy contains more than one star.

That's a fine trick for developing a simple understanding of infinities, but then apparently in later maths it completely falls over.

For example, I'm one of the many people who have a rather primal loathing for the notion that .999 repeating is really equal to 1. (I literally do have a powerful emotional response to this. I hate it. I know it's been proven. I know it's a fact. But I hate it.)

[1]: http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp...

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