lots of nothing
Not posted in a while. I've been messing about with telepathy, Windows vista and kde4 (details can be found on my wiki at http://www.srcf.ucam.org/~dl325/moin/moin.cgi/RecentChanges ), and not dancing and not doing any work. I feel that dancing and doing work may be linked. I always feel productive when I've been dancing and gotten a good night's sleep.

I've also been going to see my nan in hospital. She had a brain tumour removed a while ago, and her mood swings somehow remind me of my mum's laptop, when it halts and catches fire every so often.

I was also thinking of something that Gavin said at past midnight sometime during term. I think it was something about not having any philosophical objection to certain hypotheses relating to the set of real numbers being impossible to prove or disprove, because the set of real numbers contains numbers that are impossible to define (if you remember the proof for the fact that the set of real numbers is uncountably infinite).

This got me to thinking whether you could produce a set of numbers that could do most of the things you normally want to do with the set of real (or complex) numbers, but is countably infinite. Let's call my set Cc (the countable set of complex numbers)

So what do I want my set of numbers to do?

1) be countable.
2) contain the rational numbers
3) satisfy f(x,y) in Cc for all x,y in Cc (for functions involving powers, like f(x,y)=x^y )
4) (for bonus marks) contain pi, and other "interesting values" like e

Is it possible to create such a set, or is there some proof that says "this is impossible"?

My first idea was to create a set of the "rational powers of rational numbers" (letting you write numbers of the form x=(i/j)^(k/l), which would be countably infinite). This doesn't even satisfy 3) though, because 2^(2^(1/2)) is an "irrational power of a rational number". My instinct says it might not even be guaranteed to satisfy 3) for functions like f(x,y)=x*y. This is Not Good.

Any mathmos fancy coming to my rescue?