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 John Conway 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, \(f_{1/2}(n)\) or \(f_{\sqrt{\Gamma_0}}(n)\) 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 • talk • contribs 20:22, March 19, 2013 (UTC)
Well, we can at extend tetration: \(a \uparrow\uparrow (1/b)\) by \(\sqrt[a \uparrow\uparrow (b-1)]{(a)}\) and pentation: \(a \uparrow\uparrow\uparrow (1/b)\) = \(\sqrt[a \uparrow\uparrow\uparrow (b-1)]{(a)}_s\). I believe that definition is reasonable, since as b tends to infinity, we get closer to 1, and for b=1 it limits to a. Ikosarakt1 (talk ^ contribs) 20:42, March 19, 2013 (UTC)
Extending tetration and the Ackermann function has been discussed to death at http://math.eretrandre.org/tetrationforum/index.php so we can probably get some useful ideas from there.
Is \(\varepsilon_\pi\) really defined? I'd like to see the definition.
Ikosarakt, what about \(a \uparrow\uparrow (b/c)\) for b > 1? Deedlit11 (talk) 21:13, March 19, 2013 (UTC)
In the matter of \(a \uparrow\uparrow (b/c)\), it would be defined as \(\sqrt[a \uparrow\uparrow (c-1)]{(a \uparrow\uparrow b)}\). It also have a good connection with integer results, for example to calculate \(3 \uparrow\uparrow (3/3)\) we need to extract \(3 \uparrow\uparrow 2\) = 27-th root of 7625597484987. The result is 3, and so even under my definition \(3 \uparrow\uparrow (3/3) = 3 \uparrow\uparrow 1\). Ikosarakt1 (talk ^ contribs) 21:24, March 19, 2013 (UTC)
Wait, that no works. Under that, \(2 \uparrow\uparrow (1/2) = 2 \uparrow\uparrow (2/3) = 2 \uparrow\uparrow (3/4)\), etc... so that problem remains undecided. Ikosarakt1 (talk ^ contribs) 21:45, March 19, 2013 (UTC)
- @Deedlit This is sort of off-topic, but if I have it correctly
\[\varepsilon_{a} = \{\varepsilon_{a^L} + 1, \omega^{\varepsilon_{a^L} + 1}, \omega^{\omega^{\varepsilon_{a^L} + 1}}, \ldots|\varepsilon_{a^R} - 1, \omega^{\varepsilon_{a^R} - 1}, \omega^{\omega^{\varepsilon_{a^R} - 1}}, \ldots\}\]
- e.g.
\[\varepsilon_{1/2} = \{\varepsilon_0 + 1, \omega^{\varepsilon_0 + 1}, \omega^{\omega^{\varepsilon_0 + 1}}, \ldots|\varepsilon_1 - 1, \omega^{\varepsilon_1 - 1}, \omega^{\omega^{\varepsilon_1 - 1}}, \ldots\}\]