User blog:Edwin Shade/Two Exceedingly Large Numbers

In Pellucidar12's 2016 October 29th blog post, he attempted to make a largest validly defined googolism, but it came to no avail. He was however very close to a method of constructing large numbers that I feel is entirely well defined, even if I have not provided a strictly formal definition of it here.

The trick is to generalize over the construction of countable ordinals themselves, and thus create a function which produces an ordinal based on the two 'seed' ordinals $$\omega$$ and $$\delta$$, such that $$\delta>\omega$$; this new ordinal is labeled $$\beta$$. $$\beta$$ is constructed from the ordinal $$\delta$$ in a process isomorphic to the construction of $$\delta$$ from $$\omega$$, or in concise, (yet somewhat vaguer), terms, $$\beta$$ is to $$\delta$$ as $$\delta$$ is to $$\omega$$.

For instance, $$epsilon_1$$ is to $$\epsilon_0$$ as $$\epsilon_0$$ is to $$\omega$$. The rest is very intuitive, and hence I do not feel merits many examples, (though if confusion should ensue I will provide as many examples as possible to remedy this).

Notate the ordinal $$\gamma$$ such that $$\gamma$$ is to $$\delta$$ as $$\delta$$ is to $$\omega$$ as $$\delta ;1$$, and in general, let the ordinal $$\delta ;n$$ refer to the ordinal $$\gamma$$ such that $$\gamma$$ is to $$\delta ;n-1$$ as $$\delta ;n-1$$ is to $$\omega$$.

My number is $$f_;{\omega}}(^{10,000}10,000)$$, employing the fast-growing hierarchy and where $$\omega_1^{CK};\omega$$ refers to a transfinite iteration of my aforementioned notation upon the Church-Kleene ordinal This number is to be known as Apollyon, to reflect it's great size, and it's transcendence of many a mathematical notations, such that it may be thought of as residing in an 'abyss' of numbers which are normally inaccessible to the average notation.

To extend the $$\gamma ;n$$ function for transfinite values of n, the ordinal $$\gamma ;\beta$$ is to be equal to $$sup\{\gamma;\delta |\delta<\beta\}$$, when $$\beta$$ is a limit ordinal, and $$\gamma ;\alpha$$ is to be equal to $$\gamma ;(\gamma ;\alpha -1)$$ when $$\alpha$$ is a successor ordinal.

The next number I will define is $$f_{\xi}(^{12}12)$$, where $$\xi$$ is the first fixed point of the equation $$\xi=\omega_1^{CK};(\xi)$$. Again, the fast-growing hierarchy and the Church-Kleene ordinal are being used. I call this number Pellucidar's Number, as without Pellucidar's blog post I most likely would not have created this one.

Any questions or comments are welcome, as I will do my best to explain any point you may find vague.