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

so this would be where aleph-0, aleph-1, et al, come from then?

I think so...not sure. I've a feeling that aleph-1 is defined as the next biggest infinity after aleph-0 (which is the size of the natural numbers), but then the undecidability of the continuum hypothesis would mean that there wouldn't be a consensus on what aleph-1 is. I could be wrong.

Hi!

Just wondering - one of my friends said something about - and I quote - "two to the aleph null". He does University level maths (I don't, obviously), and because we were at a party it was kind of hard to see what he meant. Does the statement actually have any meaning, or did I hear him wrong? (He's gone back to his University, so I can't ask him any more )
By the way, the article is great. I was expecting to find something on Cantor's 'diagonal proof', but this was much more satisfying .

Thanks,

Jordan

2^(aleph_null) is the cardinality of the number of ways of making aleph_null choices between two things. This makes sense, if you have 3 choices to make between 2 things then you have 8 (2^3) ways to do it.

So 2^(aleph_null) is the number of functions from natural numbers to the set {0,1}, it is also the size of P(N), and turns out to be the number of real numbers.

Incidentally, since my reply above I've learnt a bit more - I think the alephs are specifically ordinals, aleph_0 is the first infinite ordinal, aleph_1 is the first uncountable ordinal.

Thanks ! It sounded as if he meant something to that effect, but I wasn't sure. I thought that maybe I had misheard.
By the way, if the alephs are ordinals, with aleph null being uncountable and aleph-one being uncountable, what is aleph-two? Or does logic break down for infinities of that order...(or analogy!)

Ok, just looked it up...the alephs are ordinals, they are the special ones that are the smallest ordinals of the cardinality that they represent. So aleph_0 is the first infinite ordinal, aleph_1 is the first uncountable ordinal. aleph_2 is the first ordinal of cardinality bigger than that of aleph_1, aleph_3 is the first ordinal with cardinality bigger than that of aleph_2, and so on...aleph_w is the first ordinal with cardinality bigger than that of any of the aleph_n for n a natural number (where w is also known as aleph_0), and so on - it carries on to aleph_aleph_1, etcetera etcetera and all gets ludicrously large

If you want to know more about Cantor's diagonal proof, look at

http://www.bbc.co.uk/h2g2/guide/A479180

Andy