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### 4.2

4.2
The Fast Growing Hierarchy
 CHAPTERSELECT1.1 Nature of Number1.2 Sys. of Numeration1.3 Arithmetic1.4 Properties1.5 Catalog of NumbersCosmic Horizons2.2 SI Prefixes2.3 Imagining Numbers2.4 The -illion Series3.1 Intro to Recursion3.2 Common Notations3.3 Large No. Arithmetic4.1 Jonathan Bowers4.2 Fast Growing4.3 Extensible-EAPPENDIXINDEX In this chapter we'll be studying in some detail the professional mathematician's standard notation for very large numbers : the Fast Growing Hierarchy (FGH). The main motivations for this are that (1) it's a system recognized and used by professional mathematicians, and therefore serves as a standard of measure for all googological functions (2) it has a very simple definition and is easier to develop than some other systems. Because of the complexity of FGH it will be broken up into several sub-systems. Very quickly we'll find that FGH surpasses anything we would encounter in Section III and has no difficultly keeping pace with BEAF!4.2.1 - Introduction to the Fast Growing Hierarchy            Here is a quick introduction into what the fast growing hierarchy is and how it came about.4.2.2 - The Fast Growing Hierarchy below ordinal ω            Our first subsystem of the Fast Growing Hierarchy is FGH_ω. This system covers primitive recursive growth rates.4.2.3 - The Fast Growing Hierarchy below ordinal ω^ω            The subsystem FGH_ω^ω has the same strength as linear arrays and xE#.4.2.4 - The Fast Growing Hierarchy below ordinal ε0            Here we examine the details of subsystem FGH_ε0. It has the same strength as tetrational arrays and my own Cascading-E Notation.4.2.5 - The Fast Growing Hierarchy below Cantor's Ordinal            Cantor's ordinal, denoted φ(2,0) using Veblen's famous notation, is our next stop! We introduce the epsilon numbers, define their fundamental sequences in a constructive manner, and then take them to the limit.4.2.6 - The Fast Growing Hierarchy below Γ(0)            Γ(0) is a well known large countable ordinal in professional mathematics. In this article we learn how to construct it using the binary Veblen function, a new concept, and then use this massive system of ordinals to construct the FGH_ Γ(0) subsystem. Many many examples are worked out, and numeric evidence suggests that BEAF goes way further than even this!