User blog comment:Edwin Shade/In The Pursuit Of Organization/@comment-1605058-20171030102151/@comment-32876686-20171030223044

I believe I've proven at least that $$a\uparrow\uparrow\infty$$ converges when $$a=e^{\frac{1}{e}}$$, as I don't see any holes in my proof, unless you could point out the mistake.

The formal definition of $$a\uparrow\uparrow\infty$$ is $$$$\lim_{n\to\infty} Z_n$$$$, where $$Z_{n+1}=a^{Z_{n}}$$ and $$Z_0=a$$.

The ellipsis at the top of the power tower just implies that the a's keep on going forever, not that there isn't a top. Just because something is infinite doesn't mean it only has one end. I'll give you an example of this.

Imagine you would like to enumerate the ancestors of a rabbit. To do so, you assign a '0' to every male, and a '1' to every female. Beginning with the rabbit's parents you can make a binary string that describes what paths to take in the family tree such that you reach the intended ancestor. So for instance, to describe the mother of the father of the father of the mother of the father of the father of a rabbit, you would use the binary string $$001001$$.

Now let's make things interesting. Let us impose the rule that the rabbits are immortal and that in the beginning, there existed an infinite number of rabbits, and that the creation of new rabbits continued indefinitely, but at an exponential pace, such that after a finite amount of time there exist rabbits with infinite ancestry.

We can choose such a rabbit, and assign a binary string to an ancestor of that rabbit infinitely far back, which has a beginning and an end. We can say even though this string is infinitely long it has both a beginning and an end, and we can assign meaningful statements to both the beginning and the end.

For instance, if the beginning digits are $$0110...$$ it means the given ancestor is an ancestor of the father of the mother of the mother of the father of the chosen rabbit, and if the ending digits are $$...1001$$ it means the ancestor is a mother of a father of a father of a mother, and so on.

Combined, we have $$0110...1001$$, which although infinite it is perfectly valid to speak of both ends. Indeed, since the rabbits are immortal the chosen rabbit would be able to meet it's infinite ancestor and shake paws with it, proof that neither the chosen rabbit nor the infinite ancestor do not exist meaningfully just because they are on both ends of an infinite string.