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The Fast Growing Hierarchy
1.1 Nature of Number

1.2 Sys. of Numeration
1.3 Arithmetic
1.4 Properties
1.5 Catalog of Numbers
2.1 Cosmic Horizons
2.2 SI Prefixes
2.3 Imagining Numbers
2.4 The -illion Series
3.1 Intro to Recursion

3.2 Common Notations

3.3 Large No. Arithmetic

4.1 Jonathan Bowers

4.2 Fast Growing

4.3 Extensible-E

            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!

            Here is a quick introduction into what the fast growing hierarchy is and how it came about.

            Our first subsystem of the Fast Growing Hierarchy is FGH_ω. This system covers primitive recursive growth rates.

            The subsystem FGH_ω^ω has the same strength as linear arrays and xE#.

            Here we examine the details of subsystem FGH_ε0. It has the same strength as 
tetrational arrays and my own Cascading-E Notation.

            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.

            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!