Forum:Breaking free from the integers

Hello, all! I'm writing to discuss a problem that's been on my mind for a long time: how can a googologist break free from the iron grip of the integers?

Daniel Geisler has done some cool things with continuous function iteration for things like tetration. I am interested in the general problem of extending googology past counting numbers. What is \(3 \uparrow\uparrow\uparrow 4.5\)? Or \(5 \uparrow^{1/2} 4\)? Or \(\{10, 100 (\pi) 2\}\)?

My math idol wrote a book discussing surreal numbers, an extension of the ordinals into things like \(\omega - 1\) and \(\varepsilon_\pi\). It might be interesting to figure out what, say, \(f_{\sqrt{\omega}}(n)\) means.

We may not be able to convert all our favorite notations into continuous forms, but there's a whole world to explore here.

FB100Z &bull; talk &bull; contribs 20:22, March 19, 2013 (UTC)