Forum:Can you help me answer my 12 year old daughters googolology question?

Hi,

Recently my 12 year old daughter was asking me about big numbers. I've always had a fascination myself - but am very much a layman by the standards of people on this site.

She knew about googolplex and wanted to know if there were bigger named numbers. I told her grahams number was much much bigger and tried to explain the construction. She got a bit lost with the multiple up-arrow notation but I wanted to encourage her interest so I set her a challenge - if she could describe a way to construct a number bigger than grahams number I would buy her a new phone! This seemed a safe bet for me as I don't think its an easy task from where she is starting - but it got her interest. Below is what she came up (she did it with pictures and hand waving - not this notation - but I've tried to formalise it for her) :

Her first idea was power towers, ie her first number was power tower of googolplexes, googolplex high, ie


 * A = googolplex ↑↑ googolplex

I explained this didn't even begin to get to the first step of grahams number. She then tried:


 * B = A ↑↑ A

Again I explained that wasn't big enough. Then she came up with a way of iterating the process :


 * f(0) = B


 * f(n) = f(n-1) ↑↑ f(n-1)

and her third number was:


 * C = f(B)

I said it still wasn't enough (though she was getting further than I'd imagined and I realised I was going to have trouble justifying that!). Finally she iterated again:


 * D = f(f(f(f(... f(C))...)  with C nested f's.

I told her I still didn't think it was big enough - but I wasn't so confident. I said it was probably bigger than G1 (3 ↑↑↑↑ 3) but still nowhere near G64?

Can anyone help me understand how far she got? I need to be able to justify to her that I don't owe her a new phone!

thanks - reddal