User:Vel!/Laws of googology

Googology is a really scattered and disorganized field, and this site is an attempt to unify it geographically. But we still haven't unified it philosophically. As far as I can tell, only Sbiis Saibian has set out to try to do this, so here's my attempt.


 * First Law of Googology: It only gets worse.
 * Corollary: You ain't seen nothin' yet!
 * The Law of Infinity: There is no infinity. Googology is a never-ending quest to outrun itself.
 * The Law of Decomposition: Every googologism (number/function/hierarchy/whatever) decomposes into lesser googologisms.
 * For example, all BEAF arrays eventually break down to the Base Rule, \(\{a, b\}\). Fast iteration hierarchies eventually come down to \(f_0(n)\). Graham's number is a power tower.
 * The Law of Non-Arbitrariness: A "good" googologism should decompose into lesser googologisms that are non-arbitrary both individually and as a whole.
 * Triakulus is a good googologism because it's the smallest non-degenerate legion array. Googol is a bad googologism, but if we consider 10 to be non-arbitrary, it's fine.
 * A salad is a googologism that decomposes into chunks that may be non-arbitrary by themselves, but make no sense when combined. BOX_M~ defines some perfectly consistent factorial extensions, but they're strung together in a completely random order.
 * The Trinity of Iterative Functions: Suppose we have an iterative googological function \(G\).
 * The Base Rule: All expressions in \(G\) must decompose to expressions in the Base Rule. (See Law of Decomposition.)
 * The Catastrophic Rule: When the Base Rule does not apply, the Catastrophic Rule carries out iterations.
 * The Prime Rule: The Prime Rule tells the Catastrophic Rule when to stop.
 * These are named exactly as in the three rules of BEAF. They are also found in FGH, although in a disguised form. It is my belief that a googological system must have these three rules in some form to get anywhere.
 * The Base Rule is the fuel, the Catastrophic Rule is the engine, and the Prime Rule is the brake. All parts of the system are necessary:
 * Without the fuel, the engine can't run.
 * Without the engine, the fuel and brake are useless.
 * Without the brake, the engine will just keep going forever (violating the law of infinity).