User blog comment:Sbiis Saibian/Introduction to Googology/@comment-168.212.150.41-20151110203629/@comment-1605058-20151110213505

If we were to be pedantic, then your expression (2^0.5)^^Infinity isn't well-defined, just like plain "Infinity" or 1/Infinity aren't. What you would have to really consider is a limit, in your example, $$\lim\limits_{n\rightarrow\infty}(2^{0.5})\uparrow\uparrow n$$. In a similar vein, Infinity should be $$\lim\limits_{n\rightarrow\infty}n$$ and 1/Infinity should be $$\lim\limits_{n\rightarrow\infty}\frac{1}{n}$$. Now all of these expressions are valid. The only problem is that the one expressing Infinity doesn't converge.