Infinity Hotel - All Rooms full but Rooms Available

[Clinton Huxley]

Reading David Deutsch's "The Beginning of Infinity" at the moment (excellent BTW) and he uses Hilbert's idea of the "Infinity Hotel" to illuminate that there are different types of infinity (Countable and unconuntable).

Hotel Infinity has an an infinite number of rooms. Each room is occupied. Yet, no matter how many new guests turn up, they can always be accommodated.

Partial explanation here:-

http://en.wikipedia.org/wiki/Hilbert's_ ... rand_Hotel

Eventually, my not-particularly mathematically-minded brain started to sulk and think about bacon but it was still interesting and made me think we could get some kind of "Infinity Server" for Rationalia, so no matter how many posts 'Zilla made, there would always be room for more...

[JimC]

I first read that particular description of infinity in Douglas Hofstadter's "Godel, Escher, Bach", which I must re-read...

I always tell my students not to treat infinity in the same way as other mathematical entities, or there will be tears before bedtime...

[Clinton Huxley]

It was the distinction between "countable" and "uncountable" infinities that made me go "Huh?"

[JimC]

It was the distinction between "countable" and "uncountable" infinities that made me go "Huh?"

If they put a "potentially" in front of those words, it might make more sense...

But it does make the brain hurt, which is exactly why I should present the concept to my advanced students. No pain, no gain...

[Clinton Huxley]

I've had Godel, Escher, Bach for years but have never got round to reading it. Recommended?

[JimC]

I've had Godel, Escher, Bach for years but have never got round to reading it. Recommended?

Absolutely!

A little eccentric, but seminal...

[Clinton Huxley]

A little eccentric, but seminal...

You could be describing yourself there, old chap

[JimC]

A little eccentric, but seminal...

You could be describing yourself there, old chap

[Hermit]

I always tell my students not to treat infinity in the same way as other mathematical entities, or there will be tears before bedtime...

...especially if you divide it by zero.

I hate it when people assume that because infinity exists in abstract/mathematical/ontological levels, it therefore also exists in the physical one.

[Xamonas Chegwé]

It was the distinction between "countable" and "uncountable" infinities that made me go "Huh?"

Countable is a misleading adjective - replace it with "orderable" and you are closer to the mark. The numbers in an orderable infinite set can always (although not always easily) be arranged in such a way that, even though there are an infinite number of them, the numbers before and after any particular number are fixed.

eg. The set of natural numbers: 1, 2, 3, 4, etc...

An uncountable set of numbers can never be so arranged. For example, there is no way to arrange the set of decimals between 0 and 0.00000001 such that there are not an uncountably infinite number of numbers between any two that you specify!

[JacksSmirkingRevenge]

'Twould be a good venue for a Ratz meet though, I feel.

[Audley Strange]

I've had Godel, Escher, Bach for years but have never got round to reading it. Recommended?

Yep. Tremendous work.