Infinity of different sizes....

[R1Geezer]

I watched one of those Maths series recently, originally on the BBC it was in three parts covering all sorts of stuff. One bit I found most interesting was the theory that you can have different size infinities. Eg there are an infinite number of integers, right? Also within each integer there are an infinite number of natural numbers, ie fractions, getting ever smaller. The theory therefore states that the infinite set of natuaral numbers is much larger than the infinite set of integers. This seems logical to me however the word "infinity" seems to not quite cover it. I tried to explain this to a mate of mine after several libations last week and made a complete hash of it! Anyone have a neat way of explaining this to a layman? Anyone know who put forward the theory to start with? thanks.

[R1Geezer]

for "natural numbers" read "real numbers", senior moment!

[jake-the-peg]

This is a subject not easilly covered in an AB post.

The first work on this that produced non-trivial work was done by Georg Cantor.. He developed Set Theory and this was an off shoot of it.

There's some stuff on Wikipedia on it but it's not particularly good - Wikipedia's not great for Maths subjects I find. They get a bit confused as to their audience and tied up in their own rigour.

It's known as Cantor's diagonal proof.

There are 5 "easy" explanations of it for here have a look and see if any of them suit.

http://www.helium.com/knowledge/128531-cantors -infinity-proof-made-easy

(Careful - Cantor ended his life in a lunatic asylum)

[Peter Pedant]

.

Chap one of Boolos and Jeffrey
Computability and Logic does it well.

Good OU course to go with it

but it is not something you should tackle with CSE Maths (even grade 1) from 1964

[R1Geezer]

thanks guys, I'll try not to end up in the looney bin myself!

[factor-fiction]

It's also interesting that things can be infinitely large or infinitely small.
If you start with number 1 you can keep doubling it (2, 4,8,16,32,...) forever and get an infinitely large number.
Or you can keep halving it (�, �, ⅛...) forever and get closer and closer to zero but never reach 0.

And fractals are fascinating- a snowflake-type shape can have an infinitely large perimeter but an infinitesemally small area.
http://educ.queensu.ca/~fmc/january2003/KochSn owflakeInfinite.html

[R1Geezer]

so you can have an infinately large perimeter and an infinately small area! My brain hurts!