User blog:DontDrinkH20/Expanded Googological Hierarchy

The Googological Hierarchy
In my previous blog post, I rattled on for several paragraphs about my "Type A, B, and C" classifications of googological specimen. In this post, I will go into detail on the expanded full hierarchy I have since completed. The motivation behind these classifications is the fact that there are obvious differences in the ways certain googologies are defined, and usually those differences can show up in their size as well.

Imaginable Googolisms
The set of imaginable googolisms is the smallest type of googolism. These specimen can be easily imagined as quantities in the brain. For example, the amount of letters in a novel is imaginable. These are usually not considered true googolisms. Here are some examples of imaginable numbers:
 * \(6\)
 * \(100\)
 * \(1532\)
 * \(2167\)

Conceivable Googolisms
The conceivable googolisms are the smallest type of googolisms which are seriously considered for their relation to googology. These specimen are at most as big as a specimen which can be easily defined with Knuth's arrow notation. Generally, they can be expressed as numbers like \(x\uparrow^y z\) for imaginable \(x\), \(y\), and \(z\). These numbers often have names similar to the name "googol". Here are some examples of (exclusively) conceivable googolisms:
 * Googol
 * Googolplex
 * Millionduplex
 * (most numbers with a similar name)
 * Hyper-E notation

Reasonable Googolisms
The reasonable googolisms, collectively known as the Type A googolisms, make up almost the entirety of the website. These specimen are easily defined (or approximated at least) using some reasonable amount of recursion on exponentiation or similar functions. It is quite likely that every number with similar naming principles to googolplex, that is, those numbers which are named by adding suffixes and prefixes to already silly/childish sounding names, are in this class. (Exclusively reasonable) examples include:


 * Almost all numbers defined using BEAF or BAN (like Meameamealokkapoowa oompa or Triakulus
 * Graham's number and similar numbers
 * \(f_{\epsilon_n}(x)\) and the most often used functions of the FGH and SGH
 * The entirety of the Golapulus group

Recursively Definite Googolisms
The recursively definite googolisms, as the name suggests, have a reasonably sized (a person could write it over the course of a week) algorithm which produces them. If they are a function, then this becomes well-defined: a googological function is Recursively definite iff it is computable. Interesting to note are the unreasonable recursively definite googolisms, which I call the Type B-recursive googolisms. Generally, they are based on combinatorical principles rather than recursion or exponentiation (though these methods could both be used). There is no obvious reason as to why they are large (in most cases, although loader's number is large for an obvious reason). Here are some examples of Type B-recursive googolisms:


 * TREE(n), SCG(n), and most of Harvey Friedman's computable googolisms
 * Every Hydra
 * Loader's number and the other entries in the Bignum contest
 * Some of the computable numbers in the combinatorics category of the wiki, like Goodstein sequences

Definite Googolisms
The definite googolisms are a generalization of the recursively definite googolisms. These specimen can be easily expressed (their existence can be formally and easily stated by a person without them giving up) using finitary arithmetic or even FOST. These googolisms, without the Type A googolisms, are called the Type B googolisms (Definite - Type A = Type B). Definite - Recursively Definite is known as Uncomputable Definite because unrecursively is not a real word. Here are some examples of uncomputable definite googolisms:


 * \(\Xi(n)\)
 * The busy beaver function \(\Sigma(n)\)
 * The doodle function doodle(n, m)
 * The betti numbers \(b_k(N)\) (which look very out of place because they seemingly don't relate to computability theory at all)

Expressible Googolisms
The expressible googolisms, which are also precisely the class of all valid googolisms, can be thought of as those which can be formalized by some person and then summarized on, say, a wikipedia page. More formally (yet still relatively subjective), these specimen can be easily defined using metamathematical logic and languages which metamathematical logic can interpret. Every valid googolism is expressible, so the exclusively expressible googolisms are precisely the valid indefinite googolisms, which pretty much have to be defined as being difficult to define. These exclusively expressive googolisms are known as Type C googolisms. Here are some examples:


 * FOST(n) (Rayo's function)
 * FOOT(n)
 * Little Bigeddon, the current largest valid googolism
 * Sasquatch (once the community gets around to trying to understand complex model theoretic and set theoretic assumptions rather than playing around with exponentiation and scientific notation)

Inexpressible Numbers
The inexpressible numbers make up almost the entirety of the number line. Statistically, if you were to pick a random natural number, there is a 100% chance it would be inexpressible. It's almost sad, seeing how the journey has been so long only to reveal that there were infinitely many more than you could ever even hope to imagine with any human tools. The inexpressible numbers which would exist if they could be well-defined, like Oblivion and Hollum's number are even more sad, considering that you know something like them does exist but you could never do any math on them, nor are they exact values.

This is one way to think about it, but really, we should be thinking about how cool this is. As humans, we have mathematically instigated the existence of \(\omega\), which contains every inexpressible number and more. We have willed an entire universe into existence so powerful that we can't even investigate what we have created - a mathematical omnipotence paradox. Set theory and googology are not humbling because there are some things which we can't see, but rather empowering because we have the power to see above them.