A Conversation for Bigger and Bigger Infinities

neat

Post 1

Martin Harper

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


neat

Post 2

HenryS

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.


neat

Post 3

Jordan

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 smiley - sadface)
By the way, the article is great. I was expecting to find something on Cantor's 'diagonal proof', but this was much more satisfying smiley - smiley.

Thanks,

Jordan


neat

Post 4

HenryS

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.


neat

Post 5

Jordan

Thanks smiley - smiley! 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!)


neat

Post 6

HenryS

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 smiley - smiley


neat

Post 7

me[Andy]g

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

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

Andy


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