CHAPTERSELECT | 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.ω Our first subsystem of the Fast Growing Hierarchy is FGH_. This system covers primitive recursive growth rates.ω The subsystem FGH_ω^ has the same strength as linear arrays and xE#.ε Here we examine the details of subsystem FGH__{0}. 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! |

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