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

What if there are an infinite number of boxes in each room and inside each box is another box and inside that box is another and so on unitl infinity?

Then you'd have to tip an infinite number of porters an infinite amount of money.

that's not infin to the power of infin - that's infinity cubed. which is... you guessed it... infinity.

The first thing that isn't a countable infinity (aleph-0) is any number >1 raised to the power of infinity - that's an uncountable infinity (or aleph-1)

What you write isn't QUITE true. The first infinity after Aleph-0 MIGHT be Aleph-1. Either that, or it might be the Continuum. Or maybe something completely different. There's no way of knowing, as the so-called Continuum hypothesis (stating that the Continuum and Aleph-0 are the same thing) has been proven to be independent of the current axioms of set theory.
In other words, you can choose to say that the hypothesis is true, or that it is false. Both are possible. The results are slightly different, though.
It is, however, true that n^(Aleph-0), where n>=2, is equal to Aleph-1. That is even how it is defined. =) In fact, that's how the whole series of Aleph cardinals is defined: Aleph-m = n^(Aleph-(m-1)).
If anybody wants to write an entry on the Aleph numbers and the Continuum hypothesis, find the project page for set theory and have a look around.

bah - I've never been a fan of this continuum idea - show me a set which is bigger than aleph-0, and smaller than aleph-1, and I'll be convinced, but until then, hypothesising about whether it *might* be possible seems pretty pointless.

Okay, here's your set: The set of all real numbers (aka the Continuum). It's bigger than Aleph-0 and smaller than Aleph-1. At least I decided right now that that's the way it is. You can choose to hold a different view. That's the whole problem with statements that are independent of the axiomatic system in use.
Another classical, and simpler for laymen to understand, example is the statement "This statement is true". Is it true or false? If you say it's true, it's true. If you say it's false, it's false. It's simply completely undeterminable.
You can start playing with concepts like "both true and false" or "neither true nor false" or strange modal logic things like "with necessity true" if you want...but I won't... *grin*

bleh - if there's something that's between a0 and a1, it ain't the set of real numbers - there's a proof that they are equivalent to the power set of the integers. Heck, I can do it for you now, should you like - only basic set theory axioms assumed.

Which set theory axioms? Standard ZFC? I doubt that, as it has now been known for almost 40 years that that CAN'T be proved. Gödel showed that the hypothesis was consistent with the ZFC system of axioms, and Cohen proved that it was undecidable.
Of course, you can always adopt another system of axioms (or add an axiom to ZFC) - there is nothing wrong with that - under which the hypothesis can either be proved or disproved, but that's an entirely different matter.

*shrug* I was just going to show one-one equivalence of the set of reals with the set of sets of integers...

Ie - every real can be represented by an (infinite) string of ones and zeros, and every string of ones and zeros can represent a real. Every string is also equal to a *list* of integers, by looking at the number of 1s between each zero: 111101101001 = {4,2,1,0,1}

The set of lists contains the set of sets. so lists>=sets. But each set can be viewed as pairs of numbers, where the first is the position in the list, and the second is the value - hence sets>=lists. so sets=lists. so sets=reals.

I skipped the 'rigour' section of the course...

It seems we're arguing about different things. *grin*
Do you want to write an entry or three for the set theory project coming up soon? More info on http://www.h2g2.com/A484625

Hmm - We'll see...
If you've got any entries where the original author has wandered off, I'd be reasonably happy to finish them off - otherwise it all depends how I feel when I hit that 'tell h2g2' button...

I have to agree with Lucinda on this one, Aleph-1 is defined as the power set of Aleph-0, and this (Aleph-1) has the same cardinality as R. The Continuum Hypothesis asks whether or not there is a set with cardinality greater than Aleph-0 and less than Aleph-1. I guess that we'll never be able to construct an example (in ZF or ZFC) of such a set, given the undecidability of CH. It raises some funny questions about what you mean by existence .