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### 4.3

4.3
Extensible-E
 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 I introduce a notation I invented as a kid, and develop it to a level comparable to many of the array-structures in BEAF. Much of this material draws on the ideas explored in Chapters 4.1 and 4.2, in particular, ordinal notations, and the idea of array-structures. 4.3.1 - A 2nd Grader's Close Encounter with the Infinite            This is the story of how I first got involved with large numbers, and the first notation I developed, referred to here as "Poly-Cell Notation". This is the notation on which the Extensible-E System is originally based! Also in this article I provide a complete, compact, and formal definition for both Hyper-E and Extended Hyper-E notation. 4.3.2 - Hyper-E Numbers (1 - 1141)             Due to the popularity of my E# numbers, I've created an expanded list of number names constructable within the system, some modifications that make it very easy to generate millions of intermediate names, and even some additional images and illustrations to help you try to comprehend the incomprehensible.4.3.3 - Extended Hyper-E Numbers (1142 - 2684)            We continue our massive list of numbers into the realms of Extended Hyper-E Notation.4.3.4 - Cascading-E Notation            In this article I establish a simple and easy way to reach epsilon-zero of the Fast Growing Hierarchy (FGH), as well as Bowers' tetrational arrays, by Extending xE# to Cascading-E Notation (E^) and Limited Extension Cascading-E Notation (LECEN). To do so I introduce a new symbol, "^", the Cascade. I go on to establish a formal definition for E^.4.3.5 - Cascading-E Numbers (2685 - 5109)             In this article I introduce a plethora of new E^ numbers. Lastly I tentatively develop LECEN and speculate about it's further extension. 4.3.6 - Extended Cascading-E Notation             In this follow up article I pick where we left off with E^ and develop a fully functional Extended Cascading-E Notation (xE^) with order type φ(ω,0,0). xE^ also goes at least as far as XX-arrays in BEAF (possibly further). A new theory of the climbing method and ordinal hyper-operations is introduced, and a new symbol, the caret top ( > ) is introduced.4.3.7 - xE^ Numbers Part I (5110 - 7399)             The first part in a massive catalog of new googolism's specifically for xE^. The list begins where we left off last with the revised Tethrathoth Regiment. We pass up LECEN and end at about E100(#^^#^6)100, which is on the order of φ(6,0).4.3.8 - xE^ Numbers Part II (7400 - 9009)            In Part II we go from φ(6,0) to φ(10,0).4.3.9 - xE^ Numbers Part III (9010 - 11870)            We finally reach the tethratope regiment with an order of φ(ω,0). After we finish up the tetrational cases we begin the Pentacthulhum Regiment with an order of φ(1,0,0) , and then the sky is the limit. By the end of the article we reach φ(1,8,0).4.3.10 - xE^ Numbers Part IV (11871 - 14070)             In Part IV we finish up the pentational cases and then move on to the hexational, then heptational cases. This Part covers orders from φ(1,8,0) to φ(4,0,0).4.3.11 - xE^ Numbers Part V (14071 - 15610)            In Part V we wrap up the xE^ Numbers at around an order of φ(6,0,0). Then I introduce the next level of development passing up φ(ω,0,0) then φ(1,0,0,0) and from there blast off towards infinity and the unknown...4.3.12 - ???????? [LOCKED]            What lies in the future of ExE ... only time will tell ...