h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

This post has been removed.

Your mistake is in assuming that the normal rules of natural numbers still apply when you're talking about infinity. The infinite Hotel Paradox attempts to show that there is a concept of infinity which you can talk about which doesn't obey the rules. If N is the total number of natural numbers, and is infinite, then N = N + 1, N = N * 2 and various other formulas which definitely do not apply to normal numbers.

OK, but then there is absolutely no sense in doing any mathematical operation with an infinite number.
More precisely, if this is the premise, then there is just one and *only one* infinite number and you cannot possibly add anything to it.
Another approach would be geometry.
A straight line continues endlessly in both directions.
You simply cannot add anything to it.
But maybe I miss the point again. Could you tell me of any practical example, where doing maths with an infinite number is actually useful? Or are there any online-resources, which could help me grasp this weird concept?
TIA

There is one simple concept of math with infinites, which is of great use: Any finite number divided by infinity is zero.

There is a slightly more advanced rule of great use, Laplace's Theorem if I remember correctly, which allows one to divide infinity by infinity and get a finite result. To put it simply, if you have a line of slope 1 and slope 2, you know any value on the second line is double that of the corresponding value on the first, even if both values are infinite.

Besides, whether or not there is practical benefit from transfinite math, it hardly matters to mathematicians.

The example with the two lines is something to think about, though I'm not 100% sure that you need different infinite numbers to cope with it.
Thanks anyway.

Here's one way to resolve your difficulty: you announce over the loudspeaker (or maybe the TV) which is installed in everyone's room that they must all pick up all their stuff and move out into the corridor immediately. When they've done this, you announce over the series of corridor-loudspeakers that they must all simultaneously walk along to the next door along. Then you get them all to enter the room whose door they are standing in front of. They all walk in and now everyone is in a room, but room 0 is empty!

*attempts to revive a dead thread*

phw: Ah, but you forget: you *can* add a finite line segment to an infinite line. Simply pick a point on the line, pull the rays apart, and splice the segment between. Bingo! Notice, however, that the line is still just as infinite.

You can add a line to a line as well. Take the lines, parallel, and divide them every, say, centimeter, and then spread the pieces by one cm. Slide the upper line over one cm and slide it down into the gaps in the lower line. One line.

That's aleph-0. Now inagine a line (segment?) with the length aleph-1. You can't begin to count aleph-1 (or so I hear, correct me if I'm wrong), so you couldn't even have a segment of an aleph-1 line. Wierd.

BTW, I've taken this entry under my wing from the Flea Market, and I'm combining it with another entry on Infinity.

-Spike A.

Hi Spike. If you are taking over the entry on countable and uncountable infiinties, you will have to be very careful, because it is easy to make mistakes when talking about these. Infinity does not respond well to common sense. In your previous posting, you talk about a line being infinite and say "That's Aleph 0". No it isn't. The number of whole numbers and the number of fractions is Aleph 0 but the number of points in the line is greater, it is an uncountable infinity.

You talk about Aleph 1. Cantor who invented Aleph 0 hoped to have a whole series of numbers Aleph 0, Aleph 1, Aleph 2 and so on. THe first one he found which is bigger than Aleph 0 is the uncountable number of points in a line. But he didn't call it Aleph 1, because there might be another one between it and Aleph 0 and it might really be Aleph 2. So right from the start, his naming system failed. Cantor called the number of points in a line C after the Continuum. So now there are two infinite numbers: Aleph 0 and C.

Well, I stand corrected!

Yeah, this one's going to take quite some research. I'd love some help from fellow math types.

-Spike A.

If you put your entry into the Writing Workshop, I'll comment on it, but it's unlikely anyone else with a Maths background will. So it is really only when it goes into Peer Review that the Maths geeks will come out in force.

I'm subscribed to this conversation and will help if you like. I don't read the writing workshop very often though. If you let me know what thread you're discussing it on, I'll subscribe to it and join in. I'm not a set theory expert, but IIRC this entry is very much on the basic side so I should be OK.