User blog:DontDrinkH20/"Interesting numbers" and the 3 Classes of Googolisms

''MORE TO BE ADDED SOON - this blog post is under construction! just wait!'' = Class I, Class II, and Class III Googolisms = There are really three classes of googolisms. There are those which are very very large but have many equivalent definitions, like TREE(3) and Graham's number, and there are others which are very very large but are defined in such a way that they don't have many equivalent definitions, such as Rayo's number, BIG FOOT, and the current largest valid googolism, Little Bigeddon.

Class I and Class III
The first class of googolisms mentioned above is what I like to call Class I, the second Class III. Class I are abundant on the wiki, by far the most often-made googolism. There are really only a few Class II googolisms (about 4 well-known ones). Class I has numbers like googolplex and Graham's number as well as functions like BEAF and FGH. Class I objects generally contain definitions of exponentiation and many nested recursive definitions. Class I googolisms can be thought of as the ones which can easily be defined in PA and where it is easy to see that they would make a very large number.

Class III googolisms on the other hand are quite scarce, although they may hold a lot of potential. Examples of Class III are Rayo's number, BIG FOOT, and the current largest valid googolism, Little Bigeddon. They typically include creating very expressive languages, and then defining numbers as "very difficult to express in those languages." There are only about 4 well-known ones. Class III googolisms can be thought of ones which are defined as being difficult to define with a certain amount of expressiveness.

The Elusive Class II
Class I is actually a bit smaller than I originally may have led you to believe, although the wiki is still mostly populated by these objects. Class I must contain objects where it is easy to see that they would produce very large numbers, typically involving repeated exponentiation and incredibly deep nested recursions. These googolisms often share equivalent values.

An example of a Class II googolism is TREE(3). However, the largest valid Class II googolism is not TREE(3). It is called SCG(n), standing for the "subcubic graph number" of n. Although it is incidentally faster growing than every single function which is provably total in a primitive theory, it is still well-defined in finitary ZF and therefore is well-defined in PA (PA and finitary ZF are mutually interpretable).

There is no obvious reason why Class II googolisms would be large at all. TREE(3) revolves around a simple game, and it seems very odd that it would create such large numbers out of practically nothing. The idea behind Class II googolisms is that they don't relate to exponentiation or recursion in any major way but still somehow manage to be googolisms.

Here are some examples:
 * TREE(n)
 * SCG(n), the subcubic graph number of n
 * The busy beaver function, \(\Sigma(n)\)
 * The Hydra function

As can be seen by the Busy Beaver function, there are some Class II googolisms which do have some intuition on why they would be large. The reason they are still not Class I is that they have little to no recursion in their definition that would make them large.

Class II is truly the most elusive class of googolisms, regardless of the fact that there are more Class II than there are Class III. These numbers inherently have a very interesting property, and yet are out of the question for being expressed in a normal human notation. Yet, because they share so many properties with Class I, they can be grouped together to form a Class called PA-friendly googolisms. These are simply the ones which are easy to express in PA or a theory interpreted by PA (like finitary ZF).

= Interesting Numbers = Sufficiently large numbers can be thought of as interesting if they can be described in many ways; that is, they have many different definitions. However, the closer \(n\) gets to \(m\), the more formulae there are of size \(n\) which approximate \(m\). For this reason, a number could also be thought of as interesting if it can be described in very concise ways relative to it's own size; that is, there are definitions of the number which are tiny in comparison to the number itself. Examples of this include:


 * Googolplex
 * TREE(3)
 * Graham's number

Wait a second... these look a lot like our old friends, the PA-friendly googolisms! Yep! Note that while these functions are examples of such previously described "interesting numbers", this is likely not anywhere close to an equivalence. That's why the "interesting numbers" and PA-friendly googolisms should not be defined to be the same; in fact, the PA-friendly googolisms shouldn't even be defined at all. They are more of an opinion than anything.

Complexity of Formulae
There is a famous way to prove that something is true for every formula of a given first-order language: induction on complexity of formulae. This is a method of induction by showing something holds for atomic formulae and then showing that when making a formula more complex in any way, is still holds for that formula. Interestingly, a ranking system of how difficult it is to produce a formula from nothing is created out of this.

Formal definition
The formal definition of the complexity of a formula \(\varphi\) in the language of PA is as follows:


 * Let \(T_0 = \{\ulcorner 0\urcorner\}\cup\{\ulcorner x_n\urcorner : n\in\omega\} \)
 * Let \(T_{n+1} = \{\ulcorner S(t)\urcorner : t\in T_n\}\cup T_n \)
 * Let \(F_0 = \{\ulcorner a = b\urcorner : a, b\in T_0\} \)
 * Let \(F_{n+1} = \{\ulcorner a = b\urcorner : a,b\in T_{n+1}\}\cup\{\ulcorner\neg\varphi\urcorner, \ulcorner\varphi\land\psi\urcorner, \ulcorner\varphi\lor\psi\urcorner, \ulcorner\forall x_m(\varphi)\urcorner, \ulcorner\exists x_m(\varphi)\urcorner : \varphi,\psi\in F_n,m\in\omega\}\cup F_n\)
 * Let the complexity of \(\varphi\) be \(\mathcal{C}(\varphi) = \min\{n : \varphi\in F_n\}\)

\(F_n\) is a "rank" of formulae. What will be shown is that every formulae in the language of PA is in \(F_n\) for some \(n\) and that every formula in \(F_{n+1}\setminus F_n\) has length at least \(n+4\). The first proof will be done with induction on complexity of formulae:


 * 1) Let \(t\) be any term in the language of PA. If \(t\) is a constant symbol or a variable symbol, then by definition \(t\in T_0\). If \(t = \ulcorner S(u)\urcorner\) for a term \(u\in T_n\), then \(t\in T_{n+1}\). Therefore, for any term \(t\) in the language of PA, there is some \(n\) such that \(t\in T_n\) by induction on terms.
 * 2) Let \(\varphi\) be an atomic formula. Because the only predicate of the language of PA is \(\ulcorner=\urcorner\), \(varphi\) is of the form \(\ulcorner a = b\urcorner\) for terms \(a\) and \(b\). By 1. these terms are both in \(T_n\) for some two (respective) \(n\). Let \(a\in T_n\) and \(b\in T_m\) where \(k=\max\{n,m\}\). Then, \(\ulcorner a = b\urcorner\in F_k\) and so \(\varphi\in F_k\).
 * 3) Let \(\varphi\) be \(\ulcorner\neg\psi\urcorner\) where \(\psi\in F_n\). Then \(\varphi\in F_{n+1}\).
 * 4) Let \(\varphi\) be \(\ulcorner\chi\land\psi\urcorner\) where \(\psi,\chi\in F_n,F_m\) and \(\max\{n,m\}=k\). Then \(\varphi\in F_{k+1}\).
 * 5) Let \(\varphi\) be \(\ulcorner\chi\lor\psi\urcorner\) where \(\psi,\chi\in F_n,F_m\) and \(\max\{n,m\}=k\). Then \(\varphi\in F_{k+1}\).
 * 6) Let \(\varphi\) be \(\ulcorner\forall x_m(\psi)\urcorner\) where \(\psi\in F_n\). Then \(\varphi\in F_{n+1}\).
 * 7) Let \(\varphi\) be \(\ulcorner\exists x_m(\psi)\urcorner\) where \(\psi\in F_n\). Then \(\varphi\in F_{n+1}\).

The induction on complexity of formulae is complete. So, for every \(\varphi\), there is an \(n\) such that \(varphi\in n\), and therefore \(\mathcal{C}\) is defined on every \(\varphi\).

Next, I will prove by induction on natural numbers that every formula in \(F_{n+1}\setminus F_n\) has length at least \(n+4\).


 * 1) \(F_1\setminus F_0\) consists of:
 * 2) Formulae which are in \(F_0\) but with some added symbols, making each such formula at least 4 symbols long
 * 3) Formulae which are \(\ulcorner a = b\urcorner\) where \(a\) and \(b\) are terms exclusively in \(T_1\) (each of those terms are of the form \(S(t)\) where \(t\) is one symbol), meaning such formulae are at least 5 symbols long
 * 4) Let every formula of \(F_n\setminus F_{n-1}\) have length at least \(n+3\). Then, \(F_{n+1}\setminus F_n\) consists of:
 * 5) Formulae which are in \(F_n\setminus F_{n-1}\) but with some added symbols, making each such formula at least \(n+4\) symbols long
 * 6) Formulae which are \(\ulcorner a = b\urcorner\) where \(a\) and \(b\) are terms exclusively in \(T_{n+1}\) (each of those terms are of the form \(S^{n+1}(t)\) where \(t\) is one symbol), meaning such formulae are at least \{2n + 3\} symbols long

Therefore, every formula in \(F_{n+1}\setminus F_n\) has length at least \(n+4\). This also proves that the length of \(varphi\) is at least \(mathcal{C}(\varphi) + 3\). This exercise is trivial and left for the reader. (kek)