tethrafact 4.3.3 - Forging Extended Cascading-E Numbers Leaked version (2014-01-18) Copyright © Sbiis Saibian 2014, posted here with permission. The following is a leaked, unfinished draft of the article "Forging Extended Cascading-E Numbers" by Sbiis Saibian. Sbiis Saibian has written a web book, "One to Infinity: A Guide to the Finite", in which he introduces and describes various notations about googology (the mathematical study of large numbers) in great detail. He has invented his own notation called the Extensible-E System (ExE). tethrafact How to dowload it? tethrafact On 2014-01-17 (or earlier), Saibian accidnetally released an article about the numbers named with the new, un-released part of ExE, called "Extended Cascading-E Notation" (xE^). A user (Googleaarex) in Googology Wiki (googology.wikia.com) created a page with an analysis of the leaked article. On 2014-01-18, Saibian's wife reported the stolen information, resulting in Googleaarex's page being taken down and Saibian's article becoming private again. Before it was taken down, I've saved the article on my computer, in plain-text. Today, I've decided to publish the article to the general public as it's the one-year anniversary of the incident. The article has been officially released at 2014-01-30, and the content has since been refined and has been split into four articles. Currently, the link to the article below leads to the first of these four articles. === Links === * The leaked article was located on: https://sites.google.com/site/largenumbers/home/4-3/xec_numbers * One to Infinity main page: https://sites.google.com/site/largenumbers/home tethrafact PasteShr tethrafact * Saibian's wife's report about the stolen info: http://googology.wikia.com/wiki/User_talk:Googleaarex/xCE_levels The permission for this post can be seen here: http://googology.wikia.com/wiki/User_talk:Sbiis_Saibian#Anniversary_of_the_xec_numbers_leak (my Wikia username is Cloudy176, with the signature "☁ I want more clouds! ⛅") =================================================================================== tethrafact How to get it for free? tethrafact Large Numbers tethrafact How to get it for free? tethrafact home‎ > ‎4.3‎ > ‎ tethrafact How to get it? tethrafact xec_numbers tethrafact How to dowload it? tethrafact 4.3.3 Forging Extended Cascading-E Numbers tethrafact How to get it? tethrafact Introduction Now that we have developed a powerful and well defined Extended Cascading-E Notation it's time to take it out for a spin ... tethrafact PasteShr tethrafact Forging xE^ Numbers tethrafact How to dowload it? tethrafact Now that I have established Extensible-E up to φ(ω,0,0) , let's forge a plethora of new googolism's in xE^ ... Ready for the next next level ... tethrafact How to get it? tethrafact Tethrathoth Group (Reloaded) tethrafact How to get it? tethrafact We can begin more or less exactly where we left off with Cascading-E Notation (E^) and Limited Extension Cascading-E Notation (LECEN), except with a greater clarity than before. Just as before we can define: tethrafact How to dowload it? tethrafact tethrathoth = E100#^^#100 The transformation laws conveniently support this definition via Law II.A under the definition of fundamental sequences. Note that Law II.A is essentially the same rule that Aarex uses at the outset of his article of Extended Cascading-E: tethrafact How to get it? tethrafact #^...^(#...#) = #^...(#^...^(#...)) ... but there are some key differences. Firstly, it only defines this sequence if the polyponent is a single hyperion. Law II.A does not apply in situations such as #^^##. Secondly, while it seems more restrictive than Aarex's rule it's actually vastly more general: tethrafact How to use it? tethrafact &(α)^..k..^(#)[1] = &α tethrafact How to dowload it? tethrafact &(α)^..k..^(#)[n] = &(α)^..k-1..^((α)^..k..^(#)[n-1]) ... here the polybase is not restricted to just a hyperion. Rather it can be any member of the set of legal delimiters, namely the set x^. Typically this will be an epsilon number, though the rules allow the possibility of any legal delimiter being used, leading to hosts of rogues such as ##^^# for example. In this case it expands as expected... tethrafact How to get it for free? tethrafact ##^^# --> ##^##^...^##^## --> ##^##^...^(##^#*##^#*...*##^#*##^#) --> ##^##^...^(##^#*##^#*...*##^#*##*##*...*##*##) tethrafact PasteShr tethrafact This is not so strange or un-intuitive and leads to a lot of potential freedom. There is a third key difference. My rule is actually a fundamental sequence, where is Aarex's rule is not. Notice this transformation is applied regardless of what the last argument actually is. The rule is a synonym substitution, not a fundamental sequence. To give an example... @100#^^###n = @100#^#^^##n = @100#^#^#^^#n tethrafact PasteShr tethrafact This same sequence is carried out regardless of what "n" actually is. Note that the rule doesn't terminate. It doesn't specify a base case, leading to the meaningless transformation #^^# = #^#^^ tethrafact How to dowload it? tethrafact Aarex does however state a base case elsewhere as #^^# = #^#. These leads to a slightly inelegant set of synonyms... #^^# = #^# tethrafact How to get it for free? tethrafact #^^## = #^#^# #^^### = #^#^#^# tethrafact How to get it? tethrafact In this instances we see clearly that we are dealing with synonyms here, not fundamental sequences, because this is not an equality of numeric expressions in ExE, but rather an equality of delimiters. The use of synonyms however is wasteful here as we can simply use #^^#[n] to represent the entire sequence Aarex is trying to describe! Another reason this does not seem like a good choice is that it breaks a nice symmetry we have used up to this point. Every time we are applying a fundamental sequence to a hyperion limit ordinal we can interpret it as substituting a numeric value for the keystone hyperion. For example... #^#[n] = #^(n) = ###...## w/n #s tethrafact How to use it? tethrafact #^##[n] = #^(#*(n)) = #^#*#^#*...*#^# w/n #^#s So note that we can interpret hyperion tetration in the following way: tethrafact How to dowload it? tethrafact #^^#[n] = #^^(n) = #^#^^(n-1) = #^#^#^^(n-2) = ... = #^#^#^...^#^# w/n #s In essence, a hyperion is just a variable container which can be replaced by any positive integer. Aarex, on the other hand, treats them as simple counters, ie: tethrafact PasteShr tethrafact # = 1 tethrafact PasteShr tethrafact ## = 2 ### = 3 tethrafact How to use it? tethrafact etc. If that were so we'd have: tethrafact How to dowload it? tethrafact #^# = # #^## = ## tethrafact How to use it? tethrafact #^### = ### etc. tethrafact PasteShr tethrafact but then hyperion exponentiation would have no effect. This usage is inconsistent with the rest of Extensible-E. Although Extensible-E conventionally avoids the mixing of numbers and hyperions, the mixing has always been implied since the inception of Cascading-E. For example ... #1 = # tethrafact How to dowload it? tethrafact #2 = ## #3 = ### tethrafact How to get it? tethrafact etc. Furthermore, it was implicit in the operation of E^ that # acted as a symbol for a general number. ie #^# really means # raised to any positive integer power. With that in mind let us proceed. tethrafact How to dowload it? tethrafact Firstly let's check to make sure that our rules do properly evaluatetethrathoth: E100#^^#100 tethrafact How to get it for free? tethrafact = E100#^^#[100]100 = E100#^(#^^#[99])100 tethrafact PasteShr tethrafact = E100#^#^(#^^#[98])100 = E100#^#^#^(#^^#[97])100 tethrafact How to get it? tethrafact ... = E100#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^ tethrafact How to use it? tethrafact #^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^# ^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^ #^#^#^#^#^#^#^#^#^#^#^#^#^(#^^#[1])100 = E100#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^ tethrafact How to dowload it? tethrafact #^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^# ^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^#^ #^#^#^#^#^#^#^#^#^#^#^#^#^#100 tethrafact How to dowload it? tethrafact As you can see it behaves as expected. The #^^# delimiter is the earliest example for the application of rule II.A. Note that Aarex defines this number as E100#^^#^#100. However according to his definitions , since #^^# = #^#, #^^## = #^#^#, #^^### = #^#^#^#, and in general #^^#n = #^^(n+1) it follows that E100#^^#^#100 = E100#^^#^#[100]100 = E100#^^#^(100)100 = E100#^^(101)100. Thus E100#^^#^#100 is actually equal to E100#^^#101 in xE^. This is actually the 101st member of the sequence {grangol,godgahlah,godgathor,godtothol,...}. I'll use a new -thol operator to allow a continuation... tethrafact How to dowload it? tethrafact tethrathothitoth = E100#^^#101 tethrathothi-dutoth = E100#^^#102 tethrafact PasteShr tethrafact tethrathothi-tritoth = E100#^^#103 tethrathothi-tertoth = E100#^^#104 tethrafact How to dowload it? tethrafact ... much much much further in the sequence we have grand tethrathoth ... E100#^^#100#2 tethrafact How to dowload it? tethrafact That's the tethrathoth member of the #^^#-sequence! As for evaluating it we first consider the rules and look for the earliest rule which applies: The Base Rule doesn't apply because there is more than 1 argument, The Decomposition Rule doesn't apply since '#' is not an element of x^_lim, the Terminating Rule doesn't apply because the last argument isn't 1, and the Expansion Rule doesn't apply because the last delimiter is '#'. Therefore the Recursive Rule must apply. Here I'll demonstrate the validity of the Rule: @m#n = @(@m#(n-1)) tethrafact How to dowload it? tethrafact Let @ = 'E100#^^#' m = 100 tethrafact How to get it? tethrafact n = 2 Therefore: tethrafact How to get it for free? tethrafact E100#^^#100#2 = E100#^^#(E100#^^#100#(2-1)) = E100#^^#(E100#^^#100#1) We again go through the rules again, but this time applying them to the nested expression: the Base Rule doesn't apply because there is more than 1 argument, the Decomposition Rule doesn't apply because '#' is not a member of x^_lim, the Terminating Rule does apply since the last argument = 1. Again I'll demonstrate the validity of the rule: tethrafact How to dowload it? tethrafact @m&1 = @m Let @ = 'E100#^^#' tethrafact PasteShr tethrafact m = 100 Therefore: tethrafact How to get it for free? tethrafact E100#^^#(E100#^^#100#1) = E100#^^#(E100#^^#100) = E100#^^#(tethrathoth) Assuming we have finished the evaluation of tethrathoth (something which is quite impossible in practice, but we can speak in hypothetical terms) the next rule to apply would be the decomposition rule since '#^^#' is a member of x^_lim. Since '#^^#' is of the general form (α)^..k..^(β) , Ruleset II applies, and since β = #, Rule II.A applies: tethrafact How to dowload it? tethrafact E100#^^#(tethrathoth) = E100#^^#[tethrathoth]100 = E100#^(#^^#[tethrathoth-1])100 tethrafact How to dowload it? tethrafact = E100#^#^(#^^#[tethrathoth-2])100 = E100#^#^#^(#^^#[tethrathoth-3])100 tethrafact How to get it for free? tethrafact ... ... ... ... (you don't even want to know how long) ... ... ... ... = E100#^#^#^#^#^#^#^#^#^ ... ... ... ... (ditto) ... ... ... ... ^#^#^#^#^#^#^#100 tethrafact How to use it? tethrafact w/tethrathoth #s As you can see the 5 rules nicely codify xE^. In fact, these rules are the 5 fundamental laws of Extensible-E. What this means is that these rules never change. As long as we can keep defining larger hyperion sets and define their fundamental sequences we can continue to expand Extensible-E. This also means that we won't really need to specify them over and over again from this point on and they can be taken as a given. tethrafact PasteShr tethrafact The ' grand ' operator should be very familiar at this point, and we can easily define grand grand tethrathoth, grand grand grand tethrathoth, etc. This is where the ' ex ' operator comes in handy. The number proceeding ' ex ' tells us how many times the following element is to be repeated. Example: tethrafact How to use it? tethrafact three-ex-grand tethrathoth = grand grand grand tethrathoth four-ex-grand tethrathoth = grand grand grand grand tethrathoth tethrafact How to dowload it? tethrafact etc. This sequence forms the tethrathoth sequence which forms the basis of the continuation into Extended Cascading-E. We can append any number of grand's beforetethrathoth that we are capable of constructing. Using the ex-operator the following googolism's can easily be defined: tethrafact How to dowload it? tethrafact hundred-ex-grand tethrathoth = E100#^^#100#101 googol-ex-grand tethrathoth = E100#^^#100#(1+E100) tethrafact How to get it? tethrafact grangol-ex-grand tethrathoth = E100#^^#100#(1+E100#100) greagol-ex-grand tethrathoth = E100#^^#100#(1+E100#100#100) tethrafact How to use it? tethrafact gigangol-ex-grand tethrathoth = E100#^^#100#(1+E100#100#100#100) gugold-ex-grand tethrathoth = E100#^^#100#(1+E100##100) tethrafact How to dowload it? tethrafact godgahlah-ex-grand tethrathoth = E100#^^#100#(1+E100#^#100) tethrathoth-ex-grand tethrathoth = E100#^^#100#(1+E100#^#100) tethrafact PasteShr tethrafact tethrathoth-ex-grand tethrathoth-ex-grand tethrathoth = E100#^^#100#(1+E100#^^#100#(1+E100#^#100)) tethrafact How to get it for free? tethrafact This is of coarse just a primitive recursive extension so let's move on. We can notate the nth member of the tethrathoth sequence as E100#^^#100#n. The 100th member of the tethrathoth sequence would be E100#^^#100#100. Notice that if we let @ = 'E100#^^#' , we can write @100#100. This looks similar to grangol (E100#100). From this I coined the name grantethrathoth from grangol + tethrathoth. The problem is that generalizing this leads to potential problems. To allow for greater and easier generalization I will introduce the carta-operator (from the greek root carta, meaning map). Let G and H be googolism's in ExE. Let H be represented by @100. Let G be represented as E%. Then define G-carta-H as @%. Here G is being mapped to H. That is, we can imagine H as the 100th member of some sequence. We can map googol to this by treating it as the 100th member of the sequence 10,100,1000,10000,...etc. This analogy allows us to map any of the googolism's to our new sequence H. Think of @n as a function. @100#100 means grangol mapped to the function @n. This makes completely routine many of the previous googolism's mentioned in the Cascading-E article. Here are some examples: grangol-carta-tethrathoth = grantethrathoth = E100#^^#100#100 tethrafact How to use it? tethrafact greagol-carta-tethrathoth = greatethrathoth = E100#^^#100#100#100 gigangol-carta-tethrathoth = gigantethrathoth = E100#^^#100#100#100#100 tethrafact PasteShr tethrafact gorgegol-carta-tethrathoth = gorgetethrathoth = E100#^^#100##5 gulgol-carta-tethrathoth = gultethrathoth = E100#^^#100##6 tethrafact PasteShr tethrafact gaspgol-carta-tethrathoth = gasptethrathoth = E100#^^#100##7 ginorgol-carta-tethrathoth = ginortethrathoth = E100#^^#100##8 tethrafact How to get it? tethrafact This is the point at which I stopped in the previous article. The main reason was that the above pattern does not make it readily apparent how gugold and those after it should be appended to tethrathoth. Aarex took this opportunity to attempt to continue the pattern. I'll offer my own continuation below: gugolda-carta-tethrathoth = E100#^^#100##100 tethrafact How to dowload it? tethrafact graatagolda-carta-tethrathoth = E100#^^#100##100#100 greegolda-carta-tethrathoth = E100#^^#100##100#100#100 tethrafact How to get it? tethrafact grinningolda-carta-tethrathoth = E100#^^#100##100#100#100#100 gugolthra-carta-tethrathoth = E100#^^#100##100##100 tethrafact How to use it? tethrafact gugoltesla-carta-tethrathoth = E100#^^#100##100##100##100 throogola-carta-tethrathoth = E100#^^#100###100 tethrafact How to use it? tethrafact tristo-throogola-carta-tethrathoth = E100#^^#100###100###100 teroogola-carta-tethrathoth = E100#^^#100####100 tethrafact PasteShr tethrafact ... godgahlah-carta-tethrathoth = E100#^^#100#^#100 tethrafact How to dowload it? tethrafact grand godgahlah-carta-tethrathoth = E100#^^#100#^#100#2 grangahlah-carta-tethrathoth = E100#^^#100#^#100#100 tethrafact How to use it? tethrafact greagahlah-carta-tethrathoth = E100#^^#100#^#100#100#100 gigangahlah-carta-tethrathoth = E100#^^#100#^#100#100#100#100 tethrafact How to get it for free? tethrafact gorgegahlah-carta-tethrathoth = E100#^^#100#^#100##5 gulgahlah-carta-tethrathoth = E100#^^#100#^#100##6 tethrafact How to use it? tethrafact gaspgahlah-carta-tethrathoth = E100#^^#100#^#100##7 ginorgahlah-carta-tethrathoth = E100#^^#100#^#100##8 tethrafact How to dowload it? tethrafact gugoldgahlah-carta-tethrathoth = E100#^^#100#^#100##100 gugolthragahlah-carta-tethrathoth = E100#^^#100#^#100##100##100 tethrafact How to use it? tethrafact gugolteslagahlah-carta-tethrathoth = E100#^^#100#^#100##100##100##100 throogahlah-carta-tethrathoth = E100#^^#100#^#100###100 tethrafact How to use it? tethrafact teroogahlah-carta-tethrathoth = E100#^^#100#^#100####100 gotrigahlah-carta-tethrathoth = E100#^^#100#^#100#^#100 tethrafact How to get it? tethrafact gotergahlah-carta-tethrathoth = E100#^^#100#^#100#^#100#^#100 ... tethrafact How to dowload it? tethrafact godgoldgahlah-carta-tethrathoth = E100#^^#100#^#*#100 tethrafact How to dowload it? tethrafact godthroogahlah-carta-tethrathoth = E100#^^#100#^#*##100 godteroogahlah-carta-tethrathoth = E100#^^#100#^#*###100 tethrafact How to dowload it? tethrafact ... deutero-godgahlah-carta-tethrathoth = E100#^^#100#^#*#^#100 tethrafact PasteShr tethrafact trito-godgahlah-carta-tethrathoth E100#^^#100#^#*#^#*#^#100 teterto-godgahlah-carta-tethrathoth = E100#^^#100#^#*#^#*#^#*#^#100 tethrafact How to use it? tethrafact ... gridgahlah-carta-tethrathoth = E100#^^#100#^##100 tethrafact How to get it? tethrafact kubikahlah-carta-tethrathoth = E100#^^#100#^###100 quarticahlah-carta-tethrathoth = E100#^^#100#^####100 tethrafact PasteShr tethrafact ... godgathor-carta-tethrathoth = E100#^^#100#^#^#100 tethrafact How to get it? tethrafact gralgathor-carta-tethrathoth = E100#^^#100#^#^##100 godtothol-carta-tethrathoth = E100#^^#100#^#^#^#100 tethrafact How to get it? tethrafact ... ... this brings us to tethrathoth-carta-tethrathoth , E100#^^#100#^^#100, also called tethratrithoth. This leads to an even faster diagonalization... tethrafact How to use it? tethrafact tethratrithoth = E100#^^#100#^^#100 tethraterthoth = E100#^^#100#^^#100#^^#100 tethrafact How to get it? tethrafact tethrapethoth = E100#^^#*#5 tethra-exthoth = E100#^^#*#6 tethrafact How to dowload it? tethrafact tethra-epthoth = E100#^^#*#7 tethra-octhoth = E100#^^#*#8 tethrafact How to use it? tethrafact ... This leads to another level of diagonalization even more ridiculous... tethrafact How to get it for free? tethrafact E100#^^#*#100 E100#^^#*##100 tethrafact How to dowload it? tethrafact E100#^^#*#^#100 E100#^^#*#^#*#100 tethrafact How to get it for free? tethrafact E100#^^#*#^#*#^#100 E100#^^#*#^##100 tethrafact How to get it? tethrafact E100#^^#*#^#^#100 E100#^^#*#^#^#^#100 tethrafact How to get it for free? tethrafact ... This eventually leads to E100#^^#*#^^#100. Here we have the hyperion-power operators, deutero, trito, teterto, etc. These take any hyper-factor, &, and the replace it with &^(n). Where deutero = 2 , trito = 3, teterto = 4, etc. tethrafact PasteShr tethrafact Thus we have: deutero-tethrathoth = E100#^^#*#^^#100 = E100(#^^#)^#2 tethrafact PasteShr tethrafact trito-tethrathoth = E100#^^#*#^^#*#^^#100 = E100(#^^#)^#3 teterto-tethrathoth = E100#^^#*#^^#*#^^#*#^^#100 = E100(#^^#)^#4 tethrafact How to get it for free? tethrafact ... hecato-tethrathoth = E100(#^^#)^#100 tethrafact How to get it? tethrafact Aarex dubs the hecato-tethrathoth the 'tethrathor'. His definition however differs significantly, so let's examine it: 'tethrathor' = E100#^(#^^(#^#/#)*#)100 tethrafact How to dowload it? tethrafact The idea here is actually quite interesting. He uses the /# to avoid the problem that occurs in the ordinal ω^(ε0+1) of offset (see article 4.2.5). We can define /# as hyperion division, something I already suggested in the previous article as a way to define E^. Here is how this would be evaluated... E100#^(#^^(#^#/#)*#)100 = E100(#^(#^^(#^#/#))^(100)100 = tethrafact How to use it? tethrafact E100(#^(#^^(#^#/#))^(99)*#^(#^^(#^#/#)100 = E100(#^(#^^(#^#/#))^(99)*#^(#^^(#^(100)/#)100 tethrafact PasteShr tethrafact = E100(#^(#^^(#^#/#))^(99)*#^(#^^(#^99))100 = E100(#^(#^^(#^#/#))^(99)*#^(#^^100)100 tethrafact PasteShr tethrafact = E100(#^(#^^(#^#/#))^(99)*#^^(101)100 As you can see there is still an offset created by this system. However, to be fair, things are functional up to this point. Interestingly enough, xE^ also can support this format, even though the rules were not designed to. It's part of the rogue set of delimiters. For example: tethrafact How to get it? tethrafact E100#^(#^^#*#)100 By Rule I.B we have that # is being raised to a successor. So following the rule we get... tethrafact How to get it for free? tethrafact E100#^(#^^#*#)100 = E100(#^(#^^#))^(100)100 = E100(#^(#^^#))^(99)*#^#^^#100 tethrafact How to use it? tethrafact = E100(#^(#^^#))^(99)*#^#^^(100)100 = E100(#^(#^^#))^(99)*#^^(101)100 tethrafact How to dowload it? tethrafact Although the rules allow for the cantor normal forms, it has the same problem as well ... it has a slight offset. This is part of why I have opted for the more natural extension of stacking epsilon numbers. I should also note that E100(#^^(101))^#100 is not the same thing as E100#^(#^^#*#)100 even if they seem like they should be the same. The reason for this is that #^^(101) is a E^ delimiter, where as #^^# is a xE^ delimiter. They behave very differently, and #^^# has a much faster growing fundamental sequence, although it takes a while to catch up. Remember that in the case of E100#^(#^^#*#)100, we are actually getting 100 iterations of #^(#^^#) not #^(#^^100). Rather, only the last one becomes #^(#^^100). It's a long time before the 2nd to last gets it's turn. By the time it does the final argument has become mind-bogglingly massive, call it X. The 2nd to last then becomes #^(#^^X) = #^^(X+1). The +1 is an additional offset. Each #^(#^^#) will add 1 to the height of the hyperion power tower. Mind you, the effect is negligible, but it does guarantee that E100(#^^#)^#100 < E100#^(#^^#*#)100 , even though ordinal arithmetic would tell us these ordinals, (#^^#)^# and #^(#^^#*#), are the same. In any case I'll appropriate this interesting case and say: tethrathor = E100#^(#^^#*#)100 tethrafact How to dowload it? tethrafact There is a way to correct this problem. Define #-1 such that #-1[n] = #[n]-1. Treating # as a indefinite value, rather than as ω, allows for much more flexibility. We can then actually define... hecato-tethrathoth = E100#^(#^^(#-1)*#)100 tethrafact PasteShr tethrafact This was the meaning Aarex was going for. I should note that this is an illegal delimiter not strictly supported by the rules of xE^. However it isn't difficult to tweak them a bit to allow such forms. Here is how this would be evaluated... tethrafact How to use it? tethrafact E100#^(#^^(#-1)*#)100 = E100(#^(#^^(#-1)))^(100)100 tethrafact How to use it? tethrafact = E100(#^(#^^(#-1))^(99)*#^(#^^(#-1))100 = E100(#^(#^^(#-1))^(99)*#^(#^^(100-1))100 tethrafact How to get it? tethrafact = E100(#^(#^^(#-1))^(99)*#^(#^^(99))100 = E100(#^(#^^(#-1))^(99)*#^^(100)100 tethrafact How to use it? tethrafact This little correction factor will propagate throughout the expression and evaluation process. Although this is the standard progression used past ε(0) in the fast growing hierarchy, I find the offset to be somewhat unnatural. This is why I have opted instead for the stacking sequence, that treats ε(0) as a new base on which to stack ordinal exponents. The two forms leads to slightly different functions unless a correction factor is implemented as above. Aarex's terms from the 'tethrathor' group to the 'todatriath' group lie below ε(1). I did not coin any googolism's in the range between hecato-tethrathoth, and Monster-Giant. Here I'll provide some terms: tethrafact How to get it? tethrafact tethrafact = E100(#^^#)^#100 (from tetration + factor because #^^# is the only factor of the hyper-product) griduutethrathoth = E100(#^^#)^##100 tethrafact How to get it? tethrafact kubicutethrathoth = E100(#^^#)^###100 quarticutethrathoth = E100(#^^#)^####100 quinticutethrathoth = E100(#^^#)^#^#5 tethrafact How to get it? tethrafact sexticutethrathoth = E100(#^^#)^#^#6 septicutethrathoth = E100(#^^#)^#^#7 tethrafact How to get it? tethrafact octicutethrathoth = E100(#^^#)^#^#8 nonicutethrathoth = E100(#^^#)^#^#9 decicutethrathoth = E100(#^^#)^#^#10 tethrafact How to get it for free? tethrafact ... centicutethrathoth = E100(#^^#)^#^#100 ... but this isn't nearly extensible enough, and we want more ... many many more intermediate terms before we get up to Monster-Giant. See if you can figure out these constructions ... tethrafact How to use it? tethrafact tethragodgathor = E100(#^^#)^#^#100 tethragralgathor = E100(#^^#)^#^##100 tethrathraelgathor = E100(#^^#)^#^###100 tethrafact How to use it? tethrafact tethraterinngathor = E100(#^^#)^#^####100 tethrapehaelgathor = E100(#^^#)^#^#^#5 tethrafact How to use it? tethrafact tethrahexaelgathor = E100(#^^#)^#^#^#6 tethraheptaelgathor = E100(#^^#)^#^#^#7 tethra-octaelgathor = E100(#^^#)^#^#^#8 tethrafact How to use it? tethrafact ... tethragodtothol = E100(#^^#)^#^#^#100 tethrafact How to get it? tethrafact tethragraltothol = E100(#^^#)^#^#^##100 tethrathraeltothol = E100(#^^#)^#^#^###100 tethraterinntothol = E100(#^^#)^#^#^####100 tethrafact How to dowload it? tethrafact tethrapehaeltothol = E100(#^^#)^#^#^#^#5 tethrahexaeltothol = E100(#^^#)^#^#^#^#6 tethrafact How to dowload it? tethrafact tethraheptaeltothol = E100(#^^#)^#^#^#^#7 tethra-octaeltothol = E100(#^^#)^#^#^#^#8 ... tethrafact How to get it for free? tethrafact tethragodtertathol = E100(#^^#)^#^#^#^#100 tethragodpeptathol = E100(#^^#)^(#^^#)5 tethragodextathol = E100(#^^#)^(#^^#)6 tethrafact How to dowload it? tethrafact tethragodeptathol = E100(#^^#)^(#^^#)7 tethragodoctathol = E100(#^^#)^(#^^#)8 ... tethrafact How to get it for free? tethrafact tethra-tethrathoth = E100(#^^#)^(#^^#)100 Now to get even further along in the progression... tethraduliath = E100(#^^#)^(#^^#)100 tethrafact How to dowload it? tethrafact (also tethrathoth-doppelex) tethradulifact = E100(#^^#)^(#^^#*#)100 tethrafact How to get it? tethrafact griduutethraduliath = E100(#^^#)^(#^^#*##)100 kubicutethraduliath = E100(#^^#)^(#^^#*###)100 tethrafact How to get it? tethrafact quarticutethraduliath = E100(#^^#)^(#^^#*####)100 quinticutethraduliath = E100(#^^#)^(#^^#*#^#)5 sexticutethraduliath = E100(#^^#)^(#^^#*#^#)6 tethrafact PasteShr tethrafact septicutethraduliath = E100(#^^#)^(#^^#*#^#)7 octicutethraduliath = E100(#^^#)^(#^^#*#^#)8 tethrafact How to get it for free? tethrafact nonicutethraduliath = E100(#^^#)^(#^^#*#^#)9 decicutethraduliath = E100(#^^#)^(#^^#*#^#)10 ... tethrafact How to use it? tethrafact tethraduli-godgathor = E100(#^^#)^(#^^#*#^#)100 tethraduli-gralgathor = E100(#^^#)^(#^^#*#^##)100 tethrafact How to dowload it? tethrafact tethraduli-thraelgathor = E100(#^^#)^(#^^#*#^###)100 tethraduli-terinngathor = E100(#^^#)^(#^^#*#^####)100 ... tethrafact How to dowload it? tethrafact tethraduli-godtothol = E100(#^^#)^(#^^#*#^#^#)100 tethraduli-godtertathol = E100(#^^#)^(#^^#*#^#^#^#)100 tethraduli-godpeptathol = E100(#^^#)^(#^^#*#^^#)5 tethrafact PasteShr tethrafact tethraduli-godextathol = E100(#^^#)^(#^^#*#^^#)6 tethraduli-godeptathol = E100(#^^#)^(#^^#*#^^#)7 tethrafact How to dowload it? tethrafact tethraduli-godoctathol = E100(#^^#)^(#^^#*#^^#)8 ... tethrathruliath = E100(#^^#)^(#^^#*#^^#)100 tethrafact How to get it? tethrafact tethrathrulifact = E100(#^^#)^(#^^#*#^^#*#)100 griduutethrathruliath = E100(#^^#)^(#^^#*#^^#*##)100 cubicutethrathruliath = E100(#^^#)^(#^^#*#^^#*###)100 tethrafact PasteShr tethrafact tethrathruliquarticulus = E100(#^^#)^(#^^#*#^^#*####)100 tethrathruliquinticulus = E100(#^^#)^(#^^#*#^^#*#^#)5 tethrafact How to get it? tethrafact tethrathrulisexticulus = E100(#^^#)^(#^^#*#^^#*#^#)6 tethrathrulisepticulus = E100(#^^#)^(#^^#*#^^#*#^#)7 tethrathruliocticulus = E100(#^^#)^(#^^#*#^^#*#^#)8 tethrafact How to dowload it? tethrafact ... tethrathruli-godgathor = E100(#^^#)^(#^^#*#^^#*#^^#)2 tethrathruli-godtothol = E100(#^^#)^(#^^#*#^^#*#^^#)3 tethrafact How to dowload it? tethrafact tethrathruli-godtertathol = E100(#^^#)^(#^^#*#^^#*#^^#)4 tethrathruli-godpeptathol = E100(#^^#)^(#^^#*#^^#*#^^#)5 tethrafact How to get it? tethrafact tethrathruli-godextathol = E100(#^^#)^(#^^#*#^^#*#^^#)6 tethrathruli-godeptathol = E100(#^^#)^(#^^#*#^^#*#^^#)7 tethrathruli-godoctathol = E100(#^^#)^(#^^#*#^^#*#^^#)8 tethrafact How to dowload it? tethrafact ... tethraterliath = E100(#^^#)^(#^^#*#^^#*#^^#)100 tethraterlichain = E100(#^^#)^(#^^#*#^^#*#^^#*#)100 tethrafact How to get it? tethrafact tethraterligridulus = E100(#^^#)^(#^^#*#^^#*#^^#*##)100 tethraterlicubiculus = E100(#^^#)^(#^^#*#^^#*#^^#*###)100 tethraterliquarticulus = E100(#^^#)^(#^^#*#^^#*#^^#*####)100 tethraterliquinticulus = E100(#^^#)^(#^^#*#^^#*#^^#*#^#)5 tethrafact How to get it? tethrafact tethraterlisexticulus = E100(#^^#)^(#^^#*#^^#*#^^#*#^#)6 tethraterlisepticulus = E100(#^^#)^(#^^#*#^^#*#^^#*#^#)7 tethraterliocticulus = E100(#^^#)^(#^^#*#^^#*#^^#*#^#)8 tethrafact How to get it? tethrafact ... tethraterli-godgathor = E100(#^^#)^(#^^#*#^^#*#^^#*#^^#)2 ... tethrafact How to get it? tethrafact tethrapepliath = E100(#^^#)^(#^^#*#^^#*#^^#*#^^#)100 tethrapeplichain = E100(#^^#)^(#^^#*#^^#*#^^#*#^^#*#)100 ... tethrafact How to use it? tethrafact tethrapepli-godgathor = E100(#^^#)^(#^^#*#^^#*#^^#*#^^#*#^^#)2 ... tethra-exliath = E100(#^^#)^(#^^#)^#5 tethrafact How to dowload it? tethrafact tethra-epliath = E100(#^^#)^(#^^#)^#6 tethra-ocliath = E100(#^^#)^(#^^#)^#7 ... tethrafact How to dowload it? tethrafact Monster-Giant = E100(#^^#)^(#^^#)^#100 = E100(#^^#)^(#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* tethrafact How to use it? tethrafact #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#*#^^#* #^^#*#^^#*#^^#*#^^#*#^^#*#^^#)100 What could be better than that? ... tethrafact How to use it? tethrafact grand Monster-Giant = E100(#^^#)^(#^^#)^#100#2 = E100(#^^#)^(#^^#)^#[Monster-Giant]100 = E100(#^^#)^(#^^#*#^^#*#^^#*#^^#* ... ... ... ... #^^#*#^^#*#^^#*#^^#)100 tethrafact How to get it for free? tethrafact w/Monster-Giant #^^#s in the exponent What's even better than that?! ... tethrafact How to dowload it? tethrafact Monster-Gigantri = E100(#^^#)^(#^^#)^#100(#^^#)^(#^^#)^#100 Monster-Gigantet = E100(#^^#)^(#^^#)^#*#4 Monster-Giganpent = E100(#^^#)^(#^^#)^#*#5 Monster-Gigantex = E100(#^^#)^(#^^#)^#*#6 Monster-Gigantep = E100(#^^#)^(#^^#)^#*#7 tethrafact PasteShr tethrafact Monster-Gigantoch = E100(#^^#)^(#^^#)^#*#8 what's even better than that?!? ... deutero-Monster-Giant = E100(#^^#)^(#^^#)^#*(#^^#)^(#^^#)^#100 tethrafact How to dowload it? tethrafact trito-Monster-Giant = E100(#^^#)^((#^^#)^#*#)3 teterto-Monster-Giant = E100(#^^#)^((#^^#)^#*#)4 ... tethrafact PasteShr tethrafact hecato-Monster-Giant = E100(#^^#)^((#^^#)^#*#)100 Here things get kinda complicated ... tethrafact How to get it for free? tethrafact E100(#^^#)^((#^^#)^#*#^^#)100 Note that this number IS NOT E100(#^^#)^(#^^#)^#101. This number is much much smaller than E100(#^^#)^((#^^#)^#*#^^#)100. Remember that only the last #^^# get's decomposed. Only once this completely decays do we get to decompose (#^^#)^#. At this point however "n", the last argument, is so unfathomably large, that comparison with the Monster-Giant is absurdly insulting... and we are just getting started with xE^ ... tethrafact How to use it? tethrafact tethra-Monster-Giant = E100(#^^#)^((#^^#)^#*#^^#)100 tethraduli-Monster-Giant = E100(#^^#)^((#^^#)^#*#^^#*#^^#)100 tethrathruli-Monster-Giant = E100(#^^#)^((#^^#)^#*#^^#*#^^#*#^^#)100 tethrafact PasteShr tethrafact tethraterli-Monster-Giant = E100(#^^#)^((#^^#)^#*(#^^#)^#)4 tethrapepli-Monster-Giant = E100(#^^#)^((#^^#)^#*(#^^#)^#)5 tethraexli-Monster-Giant = E100(#^^#)^((#^^#)^#*(#^^#)^#)6 tethrafact How to dowload it? tethrafact tethraepli-Monster-Giant = E100(#^^#)^((#^^#)^#*(#^^#)^#)7 tethraocli-Monster-Giant = E100(#^^#)^((#^^#)^#*(#^^#)^#)8 Monster Monster-Giant = E100(#^^#)^((#^^#)^#*(#^^#)^#)100 tethrafact How to use it? tethrafact alternatively ... two-ex-Monster-Giant = E100(#^^#)^(#^^#)^##2 three-ex-Monster-Giant = E100(#^^#)^(#^^#)^##3 tethrafact How to get it for free? tethrafact ... hundred-ex-Monster-Giant = E100(#^^#)^(#^^#)^##100 tethrafact PasteShr tethrafact Monster-Giant-ex-Monster-Giant = E100(#^^#)^(#^^#)^##[Monster-Giant]100 grand hundred-ex-Monster-Giant = E100(#^^#)^(#^^#)^##100#2 = tethrafact How to use it? tethrafact E100(#^^#)^(#^^#)^##[hundred-ex-Monster-Giant]100 With that exhausted we need something new ... Monster-Grid = E100(#^^#)^(#^^#)^##100 tethrafact How to get it for free? tethrafact Monster-Cube = E100(#^^#)^(#^^#)^###100 Monster-Tesseract = E100(#^^#)^(#^^#)^####100 tethrafact How to dowload it? tethrafact ... Monster-Hecateract = E100(#^^#)^(#^^#)^#^#100 = E100(#^^#)^(#^^#)^########################################### tethrafact How to use it? tethrafact #########################################################100 Diagonalizing on a larger scale we can say... Super Monster-Giant = E100(#^^#)^(#^^#)^(#^^#)^#100 tethrafact How to use it? tethrafact Super Monster-Grid = E100(#^^#)^(#^^#)^(#^^#)^##100 Super Monster-Cube = E100(#^^#)^(#^^#)^(#^^#)^###100 tethrafact How to dowload it? tethrafact Super Monster-Tesseract = E100(#^^#)^(#^^#)^(#^^#)^####100 ... Super Monster-Hecateract = E100(#^^#)^(#^^#)^(#^^#)^#^#100 tethrafact How to get it? tethrafact ... Double-Super Monster-Giant = E100(#^^#)^(#^^#)^(#^^#)^(#^^#)^#100 tethrafact PasteShr tethrafact Triple-Super Monster-Giant = E100(#^^#)^(#^^#)^(#^^#)^(#^^#)^(#^^#)^#100 ... Here is another diagonalization series that goes even further... tethrafact How to get it for free? tethrafact tethrathoth-doppelux = E100(#^^#)^(#^^#)100 tethrathoth-treppelux = E100(#^^#)^(#^^#)^(#^^#)100 tethrafact How to use it? tethrafact tethrathoth-quadrupelux = E100(#^^#)^(#^^#)^(#^^#)^(#^^#)100 tethrathoth-quintupelux = E100(#^^#)^^#5 tethrathoth-sextupelux = E100(#^^#)^^#6 tethrafact PasteShr tethrafact tethrathoth-septupelux = E100(#^^#)^^#7 tethrathoth-octupelux = E100(#^^#)^^#8 tethrafact How to dowload it? tethrafact ... terrible tethrathoth = E100(#^^#)^^#100 = E100(#^^#)^(#^^#)^(#^^#)^ ... ... ^(#^^#)^(#^^#)^(#^^#)100 w/100 #^^#s tethrafact How to dowload it? tethrafact terrible tethrathoth-doppelux = E100((#^^#)^^#)^((#^^#)^^#)100 terrible tethrathoth-treppelux = E100((#^^#)^^#)^((#^^#)^^#)^((#^^#)^^#)100 tethrafact How to get it for free? tethrafact terrible tethrathoth-quadrupelux = E100((#^^#)^^#)^^#4 terrible tethrathoth-quintupelux = E100((#^^#)^^#)^^#5 terrible tethrathoth-sextupelux = E100((#^^#)^^#)^^#6 tethrafact How to use it? tethrafact terrible tethrathoth-septupelux = E100((#^^#)^^#)^^#7 terrible tethrathoth-octupelux = E100((#^^#)^^#)^^#8 tethrafact How to get it? tethrafact ... double-terrible tethrathoth = E100((#^^#)^^#)^^#100 double-terrible tethrathoth-doppelux = E100(((#^^#)^^#)^^#)^(((#^^#)^^#)^^#)100 tethrafact How to dowload it? tethrafact double-terrible tethrathoth-treppelux = E100(((#^^#)^^#)^^#)^^#3 double-terrible tethrathoth-quadrupelux = E100(((#^^#)^^#)^^#)^^#4 tethrafact How to dowload it? tethrafact ... triple-terrible tethrathoth = E100(((#^^#)^^#)^^#)^^#100 ... tethrafact How to get it for free? tethrafact Tethriterator Group (also Great and Terrible Tethrathoth) tethrafact PasteShr tethrafact ... at ninety-nine-ex-terrible tethrathoth we reach the tethrathoth ba'al. This is where I reached the end of LECEN previously and where the set of delimiters reached its breaking point. However we now have a complete system for continuing through all the hyper-operations. We simply use a carrot top to denote the number of applications of the '^^#' operator to '#'. I will also rechrisen this number in order for it to be more extensible. Here is the new definition and name... tethriterator = E100#^^#>#100 (also tethrathoth ba'al) tethrafact How to get it for free? tethrafact This number is now called the tethriterator (tetration + iterator), since it's delimiter iteratesthe smallest tetrational operator. With this new name it is now trivial to continue... grand tethriterator = E100#^^#>#100#2 (also Great and Terrible tethrathoth) tethrafact How to get it for free? tethrafact double grand tethriterator = E100#^^#>#100#3 triple grand tethriterator = E100#^^#>#100#4 tethrafact How to use it? tethrafact quadruple grand tethriterator = E100#^^#>#100#5 quintuple grand tethriterator = E100#^^#>#100#6 sextuple grand tethriterator = E100#^^#>#100#7 tethrafact How to use it? tethrafact septuple grand tethriterator = E100#^^#>#100#8 octuple grand tethriterator = E100#^^#>#100#9 tethrafact How to get it? tethrafact Aarex extended this series up to the 7th member, sextuple grand tethriterator, but this series is much more logically constructed and far easier to extend. It's also a trivial extension in any case as it is just the tethriterator sequence. We can easily jump to the hundred and first member of this sequence, and there are now a few ways to construct a name for it... centuple grand tethriterator = E100#^^#>#100#101 In fact the -carta- operator can easily be used to extend much further and more generally. In fact we can quickly beat anything using tethrathoth, along with the terrible- and -ex- operators. Here are some examples of how our notation easily overcomes this now: tethrafact How to use it? tethrafact tethrathoth = E100#^^#>#1 = E100#^^#100 terrible tethrathoth = E100#^^#>#2 = E100(#^^#)^^#100 tethrafact How to get it for free? tethrafact terrible terrible tethrathoth = E100#^^#>#3 = E100(#^^#)^^#)^^#100 three-ex-terrible tethrathoth = E100#^^#>#4 ... tethrafact PasteShr tethrafact hundred-ex-terrible tethrathoth = E100#^^#>#101 googol-ex-terrible tethrathoth = E100#^^#>#(1+E100) tethrafact PasteShr tethrafact ... tethrathoth-ex-terrible tethrathoth = E100#^^#>#(1+E100#^^#100) (~ E100#^^#>#100#2) tethrafact How to dowload it? tethrafact tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth = E100#^^#>#(1+E100#^^#>#(1+E100#^^#100)) (~E100#^^#>#100#3) tethrafact How to get it? tethrafact ... the largest extension using this idea is found on my Ultimate Large Number List Part II. Namely ... tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth-ex-terrible tethrathoth tethrafact How to get it? tethrafact ... This amounts to exactly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tethrafact How to dowload it? tethrafact ... this precise value does not admit of a compact form even in extended Cascading-E Notation (xE^), yet it does not extend LECEN as far as it might at first seem. All those +1s barely account for anything. One can delete all of them, and replace only the deepest nested one with +2, and still get a phenomenally larger value... E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(E100#^^#>#(2+E100#^^#>#100))))))))))))))))))))))))))))))))))))))))))))))))))))) tethrafact How to dowload it? tethrafact ... this expression can be made more compact, using the rules of xE^... E100#^^#>#(2+E100#^^#>#100)#53 tethrafact How to dowload it? tethrafact ... this is still smaller than replacing 2+E100#^^#>#100 with E100#^^#>#100#2, which is much much larger... E100#^^#>#(E100#^^#>#100#2)#53 tethrafact How to dowload it? tethrafact ... this in turn can be simplified further... E100#^^#>#(E100#^^#>#100#2)#53 tethrafact How to use it? tethrafact = E100#^^#>#(E100#^^#>#(E100#^^#>#100))#53 = E100#^^#>#100#55 tethrafact How to get it? tethrafact ... as you can see, nesting the "-ex-terrible" operator barely registers in the grander scheme of xE^. This only accounts for a single additional level of primitive recursion.grangol-carta-tethriterator (E100#^^#>#100#100), is larger than even this, and grangol-carta-tethriterator pales in comparison to what comes next... tethrafact How to dowload it? tethrafact greagol-carta-tethriterator = E100#^^#>#100#100#100 ... this would be like having a function which takes tethrathoth-ex-terrible tethrathoth-ex-terrible ... -ex-terrible tethrathoth, and then returned an expression with THAT many occurences of ex-terrible. Now repeat that a hundred times. This is only two levels of primitive recursion above LECEN. Going much further still we have ... gigangol-carta-tethriterator = E100#^^#>#100#100#100#100 tethrafact How to dowload it? tethrafact ... gugolda-carta-tethriterator = E100#^^#>#100##100 tethrafact How to get it? tethrafact ... godgahlah-carta-tethriterator = E100#^^#>#100#^#100 grand godgahlah-carta-tethriterator = E100#^^#>#100#^#100#2 tethrafact How to get it for free? tethrafact = E100#^^#>#100#^#(godgahlah-carta-tehtriterator) ... tethrafact How to get it? tethrafact gridgahlah-carta-tethriterator = E100#^^#>#100#^##100 ... godgathor-carta-tethriterator = E100#^^#>#100#^#^#100 tethrafact How to use it? tethrafact ... gralgathor-carta-tethriterator = E100#^^#>#100#^#^##100 tethrafact PasteShr tethrafact ... tethrathoth-carta-tethriterator = E100#^^#>#100#^^#100 ... tethrafact How to get it? tethrafact tethrathoth-dopplux-carta-tethriterator = E100#^^#>#100(#^^#)^(#^^#)100 Monster-Giant-carta-tethriterator = E100#^^#>#100(#^^#)^(#^^#)^#100 tethrafact PasteShr tethrafact tethrathoth-trepplux-carta-tethriterator = E100#^^#>#100(#^^#)^^#3 tethrathoth-quadruplux-carta-tethriterator = E100#^^#>#100(#^^#)^^#4 ... tethrafact How to dowload it? tethrafact terrible tethrathoth-carta-tethriterator = E100#^^#>#100(#^^#)^^#100 tethrafact PasteShr tethrafact terrible terrible tethrathoth-carta-tethriterator = E100#^^#>#100((#^^#)^^#)^^#100 ... tethrafact How to use it? tethrafact .... this brings us to tethriterator-carta-tethriterator or ... tethritertri = E100#^^#>#100#^^#>#100 tethrafact How to get it for free? tethrafact tethritertet = E100#^^#>#100#^^#>#100#^^#>#100 = E100#^^#>#*#4 tethriterpent = E100#^^#>#*#5 tethrafact How to get it for free? tethrafact tethriterhex = E100#^^#>#*#6 tethriterhept = E100#^^#>#*#7 tethrafact How to use it? tethrafact tethriteroct = E100#^^#>#*#8 tethriterenn = E100#^^#>#*#9 tethrafact PasteShr tethrafact tethriterdeck = E100#^^#>#*#10 ... tethrafact How to dowload it? tethrafact tethriterhecate = E100#^^#>#*#100 (also tethritera-by-hyperion) ... this brings us to a whole new level... tethrafact How to dowload it? tethrafact tethritera-by-deutero-hyperion = E100#^^#>#*##100 tethrafact How to use it? tethrafact tethritera-by-trito-hyperion = E100#^^#>#*###100 tethritera-by-teterto-hyperion = E100#^^#>#*####100 tethrafact How to get it for free? tethrafact tethritera-by-pepto-hyperion = E100#^^#>#*#^#5 tethritera-by-exto-hyperion = E100#^^#>#*#^#6 tethrafact How to get it for free? tethrafact tethritera-by-epto-hyperion = E100#^^#>#*#^#7 tethritera-by-ogdo-hyperion = E100#^^#>#*#^#8 tethrafact How to dowload it? tethrafact tethritera-by-ento-hyperion = E100#^^#>#*#^#9 tethritera-by-dekato-hyperion = E100#^^#>#*#^#10 tethrafact How to get it for free? tethrafact ... tethritera-by-hyperada-hyperion = E100#^^#>#*#^#100 tethrafact How to dowload it? tethrafact tethritera-by-hyperada-hyperion-by-hyperion = E100#^^#>#*#^#*#100 tethritera-by-hyperada-hyperion-by-deutero-hyperion = E100#^^#>#*#^#*##100 tethrafact How to get it for free? tethrafact tethritera-by-hyperada-hyperion-by-trito-hyperion = E100#^^#>#*#^#*###100 tethritera-by-hyperada-hyperion-by-teterto-hyperion = E100#^^#>#*#^#*####100 tethrafact How to get it? tethrafact ... tethritera-by-hyperada-hyperion-by-hyperada-hyperion = E100#^^#>#*#^#*#^#100 tethrafact How to get it? tethrafact ... tethritera-by-hyperada-deutero-hyperion = E100#^^#>#*#^##100 tethrafact How to dowload it? tethrafact tethritera-by-hyperada-trito-hyperion = E100#^^#>#*#^###100 tethritera-by-hyperada-teterto-hyperion = E100#^^#>#*#^####100 tethrafact How to dowload it? tethrafact tethritera-by-hyperada-pepto-hyperion = E100#^^#>#*#^#^#5 tethritera-by-hyperada-exto-hyperion = E100#^^#>#*#^#^#6 tethrafact PasteShr tethrafact tethritera-by-hyperada-epto-hyperion = E100#^^#>#*#^#^#7 tethritera-by-hyperada-ogdo-hyperion = E100#^^#>#*#^#^#8 tethrafact How to get it for free? tethrafact tethritera-by-hyperada-ento-hyperion = E100#^^#>#*#^#^#9 tethritera-by-hyperada-dekato-hyperion = E100#^^#>#*#^#^#10 tethrafact How to dowload it? tethrafact ... tethritera-by-hyperada-hyperada-hyperion = E100#^^#>#*#^#^#100 (also tethritera-by-hyperada-hecato-hyperion) tethrafact How to get it for free? tethrafact (also tethritera-by-hyper-tetradatri) tethritera-by-hyper-tetradatet = E100#^^#>#*#^#^#^#100 tethrafact How to get it? tethrafact tethritera-by-hyper-tetradapent = E100#^^#>#*#^^#5 tethritera-by-hyper-tetradahex = E100#^^#>#*#^^#6 tethrafact How to get it? tethrafact tethritera-by-hyper-tetradahept = E100#^^#>#*#^^#7 tethritera-by-hyper-tetrada-oct = E100#^^#>#*#^^#8 tethrafact How to get it? tethrafact tethritera-by-hyper-tetrada-enn = E100#^^#>#*#^^#9 tethritera-by-hyper-tetrada-deck = E100#^^#>#*#^^#10 tethrafact How to get it for free? tethrafact ... tethritera-by-tethrathoth = E100#^^#>#*#^^#100 (also tethritera-by-hyper-tetrada-hecate) tethrafact How to get it? tethrafact tethritera-by-terrible-tethrathoth = E100#^^#>#*(#^^#)^^#100 tethritera-by-terrible-terrible-tethrathoth = E100#^^#>#*((#^^#)^^#)^^#100 tethrafact How to use it? tethrafact tethritera-by-triterrible-tethrathoth = E100#^^#>#*(((#^^#)^^#)^^#)^^#100 tethritera-by-tetraterrible-tethrathoth = E100#^^#>#*#^^#>#5 tethrafact How to dowload it? tethrafact tethritera-by-pentaterrible-tethrathoth = E100#^^#>#*#^^#>#6 tethritera-by-hexaterrible-tethrathoth = E100#^^#>#*#^^#>#7 tethrafact PasteShr tethrafact tethritera-by-heptaterrible-tethrathoth = E100#^^#>#*#^^#>#8 tethritera-by-octaterrible-tethrathoth = E100#^^#>#*#^^#>#9 tethrafact How to get it for free? tethrafact tethritera-by-ennaterrible-tethrathoth = E100#^^#>#*#^^#>#10 tethritera-by-dekaterrible-tethrathoth = E100#^^#>#*#^^#>#11 tethrafact How to get it for free? tethrafact ... deutero-tethriterator = E100#^^#>#*#^^#>#100 tethrafact PasteShr tethrafact (also tethritera-tethriterator) (also tethritera-by-hectaterrible-tethrathoth) ... this brings us to yet another higher level of diagonalization, still within thetethriterator group... tethrafact How to dowload it? tethrafact trito-tethriterator = E100#^^#>#*#^^#>#*#^^#>#100 teterto-tethriterator = E100#^^#>#*#^^#>#*#^^#>#*#^^#>#100 tethrafact How to get it? tethrafact pepto-tethriterator = E100(#^^#>#)^#5 exto-tethriterator = E100(#^^#>#)^#6 tethrafact How to use it? tethrafact epto-tethriterator = E100(#^^#>#)^#7 ogdo-tethriterator = E100(#^^#>#)^#8 tethrafact How to get it? tethrafact ento-tethriterator = E100(#^^#>#)^#9 dekato-tethriterator = E100(#^^#>#)^#10 tethrafact How to use it? tethrafact ... isosto-tethriterator = E100(#^^#>#)^#20 tethrafact PasteShr tethrafact ... hecato-tethriterator = E100(#^^#>#)^#100 tethrafact How to use it? tethrafact ... next an even faster sequence ... tethriterchain = E100(#^^#>#)^#100 tethrafact How to get it? tethrafact tethritergrid = E100(#^^#>#)^##100 tethritercube = E100(#^^#>#)^###100 tethrafact How to get it? tethrafact tethriterquarticate = E100(#^^#>#)^####100 tethriterquinticate = E100(#^^#>#)^#^#5 tethrafact How to get it for free? tethrafact tethritersexticate = E100(#^^#>#)^#^#6 tethritersepticate = E100(#^^#>#)^#^#7 tethrafact How to get it? tethrafact tethriterocticate = E100(#^^#>#)^#^#8 tethriternonicate = E100(#^^#>#)^#^#9 tethrafact PasteShr tethrafact tethriterdecicate = E100(#^^#>#)^#^#10 ... tethrafact How to dowload it? tethrafact tethriterviginticate = E100(#^^#>#)^#^#20 tethritertriginticate = E100(#^^#>#)^#^#30 tethrafact How to get it? tethrafact tethriterquadraginticate = E100(#^^#>#)^#^#40 tethriterquinquaginticate = E100(#^^#>#)^#^#50 tethrafact PasteShr tethrafact tethritersexaginticate = E100(#^^#>#)^#^#60 tethriterseptuaginticate = E100(#^^#>#)^#^#70 tethrafact How to dowload it? tethrafact tethriteroctoginticate = E100(#^^#>#)^#^#80 tethriternonaginticate = E100(#^^#>#)^#^#90 tethrafact How to dowload it? tethrafact tethritercenticate = E100(#^^#>#)^#^#100 ... next on to even more gargantuan numbers ... tethrafact How to get it for free? tethrafact tethritergodgathor = E100(#^^#>#)^#^#100 tethrafact How to dowload it? tethrafact tethritergralgathor = E100(#^^#>#)^#^##100 tethritergodtothol = E100(#^^#>#)^#^#^#100 tethritertethrathoth = E100(#^^#>#)^#^^#100 tethrafact PasteShr tethrafact tethriter-Monster-Giant = E100(#^^#>#)^(#^^#)^#100 tethriter-terri-tethrathoth = E100(#^^#>#)^(#^^#)^^#100 tethriter-duterri-tethrathoth = E100(#^^#>#)^((#^^#)^^#)^^#100 tethriter-triterri-tethrathoth = E100(#^^#>#)^(#^^#>#)4 tethriterator-dopplux = E100(#^^#>#)^(#^^#>#)100 tethrafact How to dowload it? tethrafact tethriterator-trepplux = E100(#^^#>#)^^#3 tethriterator-quadrupplux = E100(#^^#>#)^^#4 terrible tethriterator = E100(#^^#>#)^^#100 terrible terrible tethriterator = E100((#^^#>#)^^#)^^#100 terrible terrible terrible tethriterator = E100(((#^^>#)^^#)^^#)^^#100 tethrafact PasteShr tethrafact terrible terrible terrible terrible tethriterator = E100((((#^^#>#)^^#)^^#)^^#)^^#100 ... Now imagine the number ... E100(((...(((#^^#>#)^^#)^^#)^^# ... ^^#)^^#)^^#100 w/100 ^^#s using xE^ we can now truncate this to... tethrafact How to get it for free? tethrafact E100#^^#>(#+#)100 This brings us to the next level of recursion. We can begin to stack larger and larger hyperion expression on top of #^^#. Here we go ... tethriditerator = E100#^^#>(#+#)100 tethritriterator = E100#^^#>(#+#+#)100 tethriquaditerator = E100#^^#>(#+#+#+#)100 tethrafact How to dowload it? tethrafact tethriquiditerator = E100#^^#>##5 tethrisiditerator = E100#^^#>##6 tethrisepiterator = E100#^^#>##7 tethriogditerator = E100#^^#>##8 tethrinoniterator = E100#^^#>##9 tethrafact How to get it? tethrafact tethrideciterator = E100#^^#>##10 ... ... even higher diagonalizations ... tethrigriditerator = E100#^^#>##100 tethricubiculator = E100#^^#>###100 tethrafact How to dowload it? tethrafact tethriquarticulator = E100#^^#>####100 tethriquinticulator = E100#^^#>#^#5 tethrisexticulator = E100#^^#>#^#6 tethrisepticulator = E100#^^#>#^#7 tethriocticulator = E100#^^#>#^#8 tethrafact How to get it for free? tethrafact tethrinoniculator = E100#^^#>#^#9 tethrideciculator = E100#^^#>#^#10 ... tethricenticulator = E100#^^#>#^#100 ... meanwhile in the next next dimension ... tethrafact How to get it for free? tethrafact tethrispatialator = E100#^^#>#^#100 tethrispatial-squarediterator = E100#^^#>(#^#*#^#)100 tethrispatial-cubiculator = E100#^^#>(#^#*#^#*#^#)100 tethrispatial-quarticulator = E100#^^#>(#^#*#^#*#^#*#^#)100 tethrispatial-quinticulator = E100#^^#>#^##5 tethrafact How to use it? tethrafact tethrispatial-sexticulator = E100#^^#>#^##6 tethrispatial-septiculator = E100#^^#>#^##7 tethrispatial-octiculator = E100#^^#>#^##8 tethrispatial-noniculator = E100#^^#>#^##9 tethrispatial-deciculator = E100#^^#>#^##10 tethrafact How to get it for free? tethrafact tethrispatial-centiculator = E100#^^#>#^##100 ... we still have quite a climb left to reach #^^## ... tethrideuterspatialator = E100#^^#>#^##100 tethritritospatialator = E100#^^#>#^###100 tethritetertospatialator = E100#^^#>#^####100 tethrafact How to get it for free? tethrafact tethripeptospatialator = E100#^^#>#^#^#5 tethri-extospatialator = E100#^^#>#^#^#6 tethri-eptospatialator = E100#^^#>#^#^#7 tethri-ogdospatialator = E100#^^#>#^#^#8 tethri-entospatialator = E100#^^#>#^#^#9 tethrafact How to use it? tethrafact tethri-dekatospatialator = E100#^^#>#^#^#10 ... tethri-isostospatialator = E100#^^#>#^#^#20 ... tethri-hecatospatialator = E100#^^#>#^#^#100 tethrafact How to use it? tethrafact ... and another even higher level of diagonalization ... tethri-superspatialator = E100#^^#>#^#^#100 tethri-quadratetratediterator = E100#^^#>#^#^#^#100 tethri-quintatetratediterator = E100#^^#>#^^#5 tethri-sextatetratediterator = E100#^^#>#^^#6 tethrafact PasteShr tethrafact tethri-septatetratediterator = E100#^^#>#^^#7 tethri-octatetratediterator = E100#^^#>#^^#8 tethri-nonatetratediterator = E100#^^#>#^^#9 tethri-decatetratediterator = E100#^^#>#^^#10 ... tethrafact How to use it? tethrafact tethri-vigintatetratediterator = E100#^^#>#^^#20 ... tethri-centatetratediterator = E100#^^#>#^^#100 In a very loose sense we can think of this as a double stack tetrational tower. ' #^^#>#^^# ' can be interpreted as #^^(#+#), and thus as the hypothetical ordinal w^^(w*2). However do not mistake this for a mere #^^200. That value can easily be defined as E100#^^#200 = E100#^^#[200]100 = E100#^^(200)100. That value however is already vastly smaller than a grand tethrathoth, which is E100#^^(tethrathoth)100. Remember that ordinal hyper-operations are far more complex than the hyper-operations applied to the integers! Tethri-centatetratediterator would first have to expand the #^^# on top of the stack. This must then decompose into iterations of the ^^# operator on yet smaller hyperials. A hyperial is an ordinal written using hyperions and hyperion operators. Let's look at some of the expansion... E100#^^#>#^^#100 = tethrafact PasteShr tethrafact E100#^^#>#^^(100)100 = E100#^^#>#^#^ ... ^#^#^#100 = E100#^^#>#^#^ ... ^#^#100100 = E100#^^#>#^#^ ... ^(#^#99*#^#99* ... #^#99*#^#99)100 = E100#^^#>#^#^ ... ^((#^#99)99*(#^#98)100)100 = tethrafact How to use it? tethrafact E100#^^#>#^#^ ... ^((#^#99)99*(#^#98)99*(#^#97)100)100 = ... = E100#^^#>#^#^ ... ^((#^#99)99*(#^#98)99* ... *(#^#2)99*(#^#)99*#^#)100 = E100#^^#>#^#^ ... ^((#^#99)99*(#^#98)99* ... *(#^#2)99*(#^#)99*#100)100 ... at this point the decomposition cascades down to the next level although much more complex than before. You'll notice that the decomposition follows very similar behavior to the goodstein sequence. The decomposition here is also identical to that of #^^# in a tethrathoth. However once this collapse reaches the form &*#, we do not simply expand using the expansion rule as in tethrathoth. Instead we aren't even done collapsing the hyperial above the stack. It proceeds as follows: tethrafact PasteShr tethrafact E100#^^#>(&*#)100 = E100#^^#>(&+&+&+ ... +&+&+&)100 w/100 &s ... now & must itself decompose into a series of additions. This continues for a very long time until we eventually reachthe form &+#. At this point we get... E100#^^#>(&+#)100 = tethrafact PasteShr tethrafact E100(#^^#>&)^^#>#100 = E100(#^^#>&)^^#>(100)100 = E100(((...(((#^^#>&)^^#)^^#)^^#) ... )^^#)^^#)^^#100 w/100 ^^#s after #^^#>& the decomposition of & temporarily halts here as the ^^# need to be resolved first. This leads to a massive power tower, which keeps growing and growing as ^^# operators are resolved. This goes on while & further expands into even more ^^# operators. Finally at the pinnacle of this unfathomably larger tower would be #^^# ... but then the real nightmare has only begun as the entire super-cascade itself made up of cascading components must very slowly decay. The process becomes hazy out there but what we do know is thateventually it must reduce to (#^^#)^& for some extremely complicated and long winded &. Please note that & is NOT a number but a hyperiate. It will also NOT be something "simple" like (#^^#)^(#^^#)^(#^^#)^...^(#^^#)^(#^^#) with a huge number of #^^#s. The delimiter we'll rather involve a lot of mixed expressions. Generally speaking things will get frighteningly complicated before they become simple and repetitive. This only happens whenever we get to apply the recursive rule. This is the rule that actually increases the size of the last argument. Of coarse the last argument will continue to return to the value of 100 because the first argument is 100. The huge value then get's stored in the delimiter by the number of repetitions until it feeds back into the last argument. The processes are far too long for us to ever have a complete picture of it. However we can at least know that it will return to certain forms which we can describe. eventually we would reach... E100#^^#X tethrafact PasteShr tethrafact where X is a tremendous value, virtually indistinguishable for mere mortals such as us fromtehtri-centatetratediterator itself! Suffice it to say that X is MUCH MUCH bigger than 200! We STILL have not reached #^^##! Let's continue. We now introduce an even higher level of diagonalization, by introducing the staculate operator ... staculated-tethrathoth = E100#^^#>#^^#100 staculated-terrible-tethrathoth = E100#^^#>(#^^#)^^#100 staculated-double-terrible-tethrathoth = E100#^^#>((#^^#)^^#)^^#100 tethrafact How to use it? tethrafact ... staculated-tethriterator = E100#^^#>#^^#>#100 note that the second caret top must be resolved before the first. Moving on we have... terrible staculated-tethriterator = E100(#^^#>#^^#>#)^^#100 double terrible staculated-tethriterator = E100((#^^#>#^^#>#)^^#)^^#100 tethrafact PasteShr tethrafact ... hundred-ex-terrible staculated-tethriterator = E100#^^#>(#^^#>#+#)100 ... E100#^^#>(#^^#>#+#^^#>#)100 staculated-deutero-tethriterator = E100#^^#>(#^^#>#*#^^#>#)100 tethrafact How to get it? tethrafact staculated-trito-tethriterator = E100#^^#>(#^^#>#)^#3 staculated-teterto-tethriterator = E100#^^#>(#^^#>#)^#4 ... staculated-hecato-tethriterator = E100#^^#>(#^^#>#)^#100 ... tethrafact How to use it? tethrafact staculated-terrible-tethriterator = E100#^^#>(#^^#>#)^^#100 staculated-double-terrible-tethriterator = E100#^^#>((#^^#>#)^^#)^^#100 ... staculated-tethriditerator = E100#^^#>#^^#>(#+#)100 staculated-tethritriterator = E100#^^#>#^^#>(#+#+#)100 tethrafact How to use it? tethrafact staculated-tethriquaditerator = E100#^^#>#^^#>(#+#+#+#)100 staculated-tethriquiditerator = E100#^^#>#^^#>##5 staculated-tethrisiditerator = E100#^^#>#^^#>##6 staculated-tethrisepiterator = E100#^^#>#^^#>##7 staculated-tethri-ogditerator = E100#^^#>#^^#>##8 tethrafact PasteShr tethrafact staculated-tethri-noniterator = E100#^^#>#^^#>##9 staculated-tethri-deciterator = E100#^^#>#^^#>##10 ... staculated-tethrigriditerator = E100#^^#>#^^#>##100 tethrafact How to get it? tethrafact staculated-tethricubiculator = E100#^^#>#^^#>###100 staculated-tethriquarticulator = E100#^^#>#^^#>####100 staculated-tethriquinticulator = E100#^^#>#^^#>#^#5 staculated-tethrisexticulator = E100#^^#>#^^#>#^#6 staculated-tethrisepticulator = E100#^^#>#^^#>#^#7 tethrafact PasteShr tethrafact staculated-tethri-octiculator = E100#^^#>#^^#>#^#8 staculated-tethrinoniculator = E100#^^#>#^^#>#^#9 staculated-tethrideciculator = E100#^^#>#^^#>#^#10 ... staculated-tethrispatialator = E100#^^#>#^^#>#^#100 tethrafact How to get it for free? tethrafact staculated-tethrideuterspatialator = E100#^^#>#^^#>#^##100 staculated-tethritritospatialator = E100#^^#>#^^#>#^###100 staculated-tethritetertospatialator = E100#^^#>#^^#>#^####100 ... staculated-tethri-superspatialator = E100#^^#>#^^#>#^#^#100 tethrafact How to dowload it? tethrafact staculated-tethriquadratetratediterator = E100#^^#>#^^#>#^#^#^#100 staculated-tethriquintatetratediterator = E100#^^#>#^^#>#^^#5 staculated-tethrisextatetratediterator = E100#^^#>#^^#>#^^#6 staculated-tethriseptatetratediterator = E100#^^#>#^^#>#^^#7 staculated-tethrioctatetratediterator = E100#^^#>#^^#>#^^#8 tethrafact PasteShr tethrafact staculated-tethrinonatetratediterator = E100#^^#>#^^#>#^^#9 staculated-tethridecatetratediterator = E100#^^#>#^^#>#^^#10 Here we reach the end of the tethriterator group. We are now in need of something more extensible... Tethrato Group Next we establish ' tethrato ' as the general root for a tetrational operator. Here are some new constructions for continuing along the main sequence of xE^: tethrafact How to get it? tethrafact troiturreted-tethrathoth = E100#^^#>#^^#>#^^#100 territethraturreted-sur-duturreted-tethrathoth = E100#^^#>#^^#>(#^^#)^^#100 tethriterata-turreted-sur-duturreted-tethrathoth = E100#^^#>#^^#>#^^#>#100 ... quatreturreted-tethrathoth = E100#^^#>#^^#>#^^#>#^^#100 tethrafact PasteShr tethrafact tethriterata-turreted-sur-troiturreted-tethrathoth = E100#^^#>#^^#>#^^#>#^^#>#100 cinqturreted-tethrathoth = E100#^^##5 sesturreted-tethrathoth = E100#^^##6 septurreted-tethrathoth = E100#^^##7 huiturreted-tethrathoth = E100#^^##8 tethrafact How to get it for free? tethrafact neufturreted-tethrathoth = E100#^^##9 dexturreted-tethrathoth = E100#^^##10 ... Berlin Wall = E100#^^##100,000,000 = E100#^^#>#^^#>#^^#>#^^#> ... ... >#^^#>#^^#>#^^#>#^^#100 tethrafact How to get it? tethrafact w/100,000,000 #^^#s ... this expression would be roughly 7 to 8 times the length of the real berlin wall which stretched for about 100 miles ... grand Berlin Wall = E100#^^##100,000,000#2 = E100#^^##(E100#^^##100,000,000) = E100#^^##(Berlin Wall) = E100#^^#>#^^#>#^^#>#^^#> ... ... >#^^#>#^^#>#^^#>#^^#100 tethrafact How to get it? tethrafact w/Berlin Wall #^^#s ... this expression ... stretches further than the diameter of the observable universe-- no scratch that, scale up from the Berlin Wall to the observable universe as many times as there are particles in the observable universe -- no scratch that, find out how many particles THAT would contain scales up that many times then find out how many particles -- ah, fuck it, we left analogies like this behind AGES ago. There is no possible way for mere mortals such as us to unpack this completely ... moving on ... Tethrasquare Group tethrafact How to get it for free? tethrafact ... at last we reach #^^##, identified at the closing of the Cascading-E article as the foundation of formulating xE^. Before we could not provide a fundamental sequence for it. Now we can say ... #^^##[n] = #^^#>#^^#>#^^#> ... >#^^#>#^^# w/n #^^#s Aarex instead of giving it a fundamental sequence treated it as a mere synonym. This makes it significantly less powerful. In any case we can now define the following googolism ... tethrasquare = E100#^^##100 tethrafact How to get it for free? tethrafact this expands to... E100#^^#>#^^#>#^^#> ... ... >#^^#>#^^#>#^^#100 w/100 #^^#s Already we have reached the ordinal ζ0 also called φ(2,0) or ε(0,1) using my Binary Epsilon function. This is the hypothetical level of Bowers' pentational arrays ... but we've only just begun with hyperion tetration! We have merely reached the 2nd tetrational operator, the ^^##-operator. With that in mind we can go much further... grand tethrasquare = E100#^^##100#2 grangol-carta-tethrasquare = E100#^^##100#100 tethrafact PasteShr tethrafact duble-tethrasquare = E100#^^##100#^^##100 treble-tethrasquare = E100#^^##100#^^##100#^^##100 quadreble-tethrasquare = E100#^^##*#5 quibble-tethrasquare = E100#^^##*#6 sextuble-tethrasquare = E100#^^##*#7 tethrafact How to dowload it? tethrafact septuble-tethrasquare = E100#^^##*#8 octuble-tethrasquare = E100#^^##*#9 nonuble-tethrasquare = E100#^^##*#10 ... deutero-tethrasquare = E100#^^##*#^^##100 tethrafact How to get it for free? tethrafact trito-tethrasquare = E100#^^##*#^^##*#^^##100 teterto-tethrasquare = E100#^^##*#^^##*#^^##*#^^##100 pepto-tethrasquare = E100(#^^##)^#5 exto-tethrasquare = E100(#^^##)^#6 epto-tethrasquare = E100(#^^##)^#7 tethrafact How to use it? tethrafact ogdo-tethrasquare = E100(#^^##)^#8 ento-tethrasquare = E100(#^^##)^#9 dekato-tethrasquare = E100(#^^##)^#10 ... isosto-tethrasquare = E100(#^^##)^#20 tethrafact How to use it? tethrafact ... hecato-tethrasquare = E100(#^^##)^#100 ... terrible tethrasquare = E100(#^^##)^^#100 double terrible tethrasquare = E100((#^^##)^^#)^^#100 tethrafact How to dowload it? tethrafact ... hundred-ex-terrible tethrasquare = E100(#^^##)^^#>#100 tethrithoth-ex-terrible tethrasquare = E100(#^^##)^^#>#(E100#^^#100) tethrasquare-ex-terrible tethrasquare = E100(#^^##)^^#>#(E100#^^##100) ... to go further we allow previous googolism's to act as operators on newer googolism's in a way that is analogous to how xE^ takes ordinals and applies previous operators to generate new ordinals ... tethrafact How to get it? tethrafact tethriterated-tethrasquare = E100(#^^##)^^#>#100 grand tethriterated-tethrasquare = E100(#^^##)^^#>#100#2 ... tethriditerated-tethrasquare = E100(#^^##)^^#>(#+#)100 tethritriterated-tethrasquare = E100(#^^##)^^#>(#+#+#)100 tethrafact How to get it? tethrafact tethriquaditerated-tethrasquare = E100(#^^##)^^#>(#+#+#+#)100 tethriquiditerated-tethrasquare = E100(#^^##)^^#>##5 tethrisiditerated-tethrasquare = E100(#^^##)^^#>##6 tethrisepiterated-tethrasquare = E100(#^^##)^^#>##7 tethri-ogditerated-tethrasquare = E100(#^^##)^^#>##8 tethrafact How to get it for free? tethrafact tethri-noniterated-tethrasquare = E100(#^^##)^^#>##9 tethri-deciterated-tethrasquare = E100(#^^##)^^#>##10 ... tethrigriditerated-tethrasquare = E100(#^^##)^^#>##100 tethricubiculated-tethrasquare = E100(#^^##)^^#>###100 tethrafact PasteShr tethrafact tethriquarticulated-tethrasquare = E100(#^^##)^^#>####100 tethriquinticulated-tethrasquare = E100(#^^##)^^#>#^#5 tethrisexticulated-tethrasquare = E100(#^^##)^^#>#^#6 tethrisepticulated-tethrasquare = E100(#^^##)^^#>#^#7 tethri-octiculated-tethrasquare = E100(#^^##)^^#>#^#8 tethrafact How to get it? tethrafact tethrinoniculated-tethrasquare = E100(#^^##)^^#>#^#9 tethrideciculated-tethrasquare = E100(#^^##)^^#>#^#10 ... tethrispatialated-tethrasquare = E100(#^^##)^^#>#^#100 tethrideuterspatialated-tethrasquare = E100(#^^##)^^#>#^##100 tethrafact How to get it? tethrafact tethritritospatialated-tethrasquare = E100(#^^##)^^#>#^###100 tethritetertospatialated-tethrasquare = E100(#^^##)^^#>#^####100 tethripeptospatialated-tethrasquare = E100(#^^##)^^#>#^#^#5 tethriextospatialated-tethrasquare = E100(#^^##)^^#>#^#^#6 tethrieptospatialated-tethrasquare = E100(#^^##)^^#>#^#^#7 tethrafact How to get it? tethrafact tethriogdospatialated-tethrasquare = E100(#^^##)^^#>#^#^#8 tethrientospatialated-tethrasquare = E100(#^^##)^^#>#^#^#9 tethridekatospatialated-tethrasquare = E100(#^^##)^^#>#^#^#10 ... tethrisuperspatialated-tethrasquare = E100(#^^##)^^#>#^#^#100 tethrafact How to dowload it? tethrafact tethriquadratetratediterated-tethrasquare = E100(#^^##)^^#>#^#^#^#100 tethriquintatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#5 tethrisextatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#6 tethriseptatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#7 tethrioctatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#8 tethrafact How to use it? tethrafact tethrinonatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#9 tethridecatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#10 ... tethricentatetratediterated-tethrasquare = E100(#^^##)^^#>#^^#100 ... here we reach the end at which this technique can be easily applied. Here I'll take a slight shift in gears to allow for better and more logical generalization... tethrafact How to get it for free? tethrafact tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#100 territethrastaculated-tethritertethrasquare = E100(#^^##)^^#>(#^^#)^^#100 ... tethriterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#100 tethriditerstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>(#+#)100 tethrafact How to dowload it? tethrafact tethritriterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>(#+#+#)100 tethriquaditerstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>(#+#+#+#)100 ... tethrigriditerstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>##100 tethrafact How to get it? tethrafact tethricubiterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>###100 tethriquartiterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>####100 ... tethrispatialiterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^#100 tethrisuperspatialiterstaculated-tethritertethrasquare tethrafact How to get it for free? tethrafact = E100(#^^##)^^#>#^^#>#^#^#100 tethriquadratetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#4 tethriquintatetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#5 tethrisextatetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#6 tethrafact How to get it? tethrafact tethriseptatetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#7 tethrioctatetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#8 tethrinonatetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#9 tethridecatetratediterstaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#10 tethrafact PasteShr tethrafact ... duturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#100 troiturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^#>#^^#>#^^#100 quatreturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##4 cinqturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##5 tethrafact How to dowload it? tethrafact sesturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##6 septurreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##7 huiturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##8 neufturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##9 dexturreted-tethrastaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##10 tethrafact How to dowload it? tethrafact ... tethrasquarestaculated-tethritertethrasquare = E100(#^^##)^^#>#^^##100 terrible tethrasquarestaculated-tethritertethrasquare =E100((#^^##)^^#>#^^##)^^#100 terrible terrible tethrasquarestaculated-tethritertethrasquare = E100(((#^^##)^^#>#^^##)^^#)^^#100 tethrafact How to get it? tethrafact ... hundred-ex-terrible tethrasquarestaculated-tethritertethrasquare = E100(#^^##)^^#>(#^^##+#)100 ... territethrasquarestaculated-tethritertethrasquare = E100(#^^##)^^#>(#^^##)^^#100 tethrafact How to use it? tethrafact ... you may notice how long and technical the names are getting. They are starting to sound more like chemical formulas, or taxonomy than numbers. The reason for this is the same reason why chemical names and taxonomy are contain complicated names. When you start to have thousands of similar objects to name, you have to resort to longer names with interchangable parts to maintain clarity. xE^ allows for far greater freedom than previous ExE systems. There are many many many inbetween expressions between theobvious ones such as #^^##, #^^###, #^^#### etc. In fact we still have a long way to go to reach #^^###. We've only just begun with the ^^##-operator ... duterritethrasquarestaculated-tethritertethrasquare = E100(#^^##)^^#>((#^^##)^^#)^^#100 tethrafact How to get it? tethrafact tethritertethrasquarestaculated-tethritertethrasquare = E100(#^^##)^^#>(#^^##)^^#100 (also duturreted-tethritertethrasquare) tethrafact How to use it? tethrafact troisturreted-tethritertethrasquare = E100(#^^##)^^#>(#^^##)^^#>(#^^##)^^#100 quatreturreted-tethritertethrasquare = E100(#^^##)^^##4 cinqturreted-tethritertethrasquare tethrafact How to dowload it? tethrafact = E100(#^^##)^^##5 sesturreted-tethritertethrasquare = E100(#^^##)^^##6 septurreted-tethritertethrasquare = E100(#^^##)^^##7 tethrafact How to get it for free? tethrafact huiturreted-tethritertethrasquare = E100(#^^##)^^##8 neufturreted-tethritertethrasquare = E100(#^^##)^^##9 dexturreted-tethritertethrasquare tethrafact How to use it? tethrafact = E100(#^^##)^^##10 ... and the nightmare continues ... and continues ... and continues ... and continues ... secundotethrated-tethrasquare = E100(#^^##)^^##100 terrible secundotethrated-tethrasquare = E100((#^^##)^^##)^^#100 tethriterated-secundotethrated-tethrasquare = E100((#^^##)^^##)^^#>#100 tethrafact PasteShr tethrafact duturreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^#)^^#>((#^^##)^^#)^^#100 troisturreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^##)^^##3 quatreturreted-tethritersecundotethrated-tethrasquare tethrafact How to dowload it? tethrafact = E100((#^^##)^^##)^^##4 cinqturreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^##)^^##5 sesturreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^##)^^##6 tethrafact How to get it for free? tethrafact septurreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^##)^^##7 huiturreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^##)^^##8 neufturreted-tethritersecundotethrated-tethrasquare tethrafact How to get it? tethrafact = E100((#^^##)^^##)^^##9 dexturreted-tethritersecundotethrated-tethrasquare = E100((#^^##)^^##)^^##10 ... thrice-tethrasecunda = E100((#^^##)^^##)^^##100 tethrafact How to get it for free? tethrafact quatrice-tethrasecunda = E100(((#^^##)^^##)^^##)^^##100 quincice-tethrasecunda = E100#^^##>#5 sextice-tethrasecunda = E100#^^##>#6 septice-tethrasecunda = E100#^^##>#7 octice-tethrasecunda = E100#^^##>#8 tethrafact How to get it for free? tethrafact nonice-tethrasecunda = E100#^^##>#9 decice-tethrasecunda = E100#^^##>#10 ... Tethrasquared-iterator Group Here we reach a familiar form ... tethrafact PasteShr tethrafact tethrasquared-iterator = E100#^^##>#100 This is basically the same as a tethriterator, except that we are now iterating the 2nd tetrational operator, the tethrasecunda. With this in mind we can construct similar googolisms for the ^^##-operator, following the pattern of the iterated ^^#-operator... tethrasquared-du-iterator = E100#^^##>(#+#)100 tethrasquared-triterator = E100#^^##>(#+#+#)100 tethrasquared-quaditerator = E100#^^##>(#+#+#+#)100 tethrafact How to get it for free? tethrafact tethrasquared-quiditerator = E100#^^##>##5 tethrasquared-siditerator = E100#^^##>##6 tethrasquared-sepiterator = E100#^^##>##7 tethrasquared-ogditerator = E100#^^##>##8 tethrasquared-noniterator = E100#^^##>##9 tethrafact How to get it for free? tethrafact tethrasquared-deciterator = E100#^^##>##10 ... tethrasquared-griditerator = E100#^^##>##100 tethrasquared-cubiculator = E100#^^##>###100 tethrasquared-quarticulator = E100#^^##>####100 tethrafact How to get it? tethrafact tethrasquared-quinticulator = E100#^^##>#^#5 tethrasquared-sexticulator = E100#^^##>#^#6 tethrasquared-septiculator = E100#^^##>#^#7 tethrasquared-octiculator = E100#^^##>#^#8 tethrasquared-noniculator = E100#^^##>#^#9 tethrafact How to use it? tethrafact tethrasquared-deciculator = E100#^^##>#^#10 ... tethrasquared-spatialator = E100#^^##>#^#100 ... keep in mind we are now iterating the ^^##-operator using #^# diagonalization! So we are moving away from tethrasquare at an alarming rate. We still have further to go however to reach the first application of the ^^###-operator... tethrasquared-superspatialator = E100#^^##>#^#^#100 tethrafact How to use it? tethrafact tethrasquared-quadratetratediterator = E100#^^##>#^#^#^#100 tethrasquared-quintatetratediterator = E100#^^##>#^^#5 tethrasquared-sextatetratediterator = E100#^^##>#^^#6 tethrasquared-septatetratediterator = E100#^^##>#^^#7 tethrasquared-octatetratediterator = E100#^^##>#^^#8 tethrafact PasteShr tethrafact tethrasquared-nonatetratediterator = E100#^^##>#^^#9 tethrasquared-decatetratediterator = E100#^^##>#^^#10 ... tethraturreted-tethrasquare = E100#^^##>#^^#100 duturretetethraturreted-tethrasquare = E100#^^##>#^^#>#^^#100 tethrafact How to get it for free? tethrafact troisturretetethraturreted-tethrasquare = E100#^^##>#^^#>#^^#>#^^#100 quatreturretetethraturreted-tethrasquare = E100#^^##>#^^##4 cinqturretetethraturreted-tethrasquare = E100#^^##>#^^##5 sesturretetethraturreted-tethrasquare = E100#^^##>#^^##6 septurretetethraturreted-tethrasquare = E100#^^##>#^^##7 tethrafact How to get it? tethrafact huiturretetethraturreted-tethrasquare = E100#^^##>#^^##8 neufturretetethraturreted-tethrasquare = E100#^^##>#^^##9 dexturretetethraturreted-tethrasquare = E100#^^##>#^^##10 ... duturreted-tethrasquare = E100#^^##>#^^##100 tethrafact How to get it for free? tethrafact terrible duturreted-tethrasquare = E100(#^^##>#^^##)^^#100 tethriterated duturreted-tethrasquare = E100(#^^##>#^^##)^^#>#100 terrisecunded duturreted-tethrasquare = E100(#^^##>#^^##)^^##100 double terrisecunded duturreted-tethrasquare = E100((#^^##>#^^##)^^##)^^##100 ... tethrafact How to get it for free? tethrafact hundred-ex-terrisecunded duturreted-tethrasquare = E100#^^##>(#^^##+#)100 ... troisturreted-tethrasquare = E100#^^##>#^^##>#^^##100 quatreturreted-tethrasquare = E100#^^##>#^^##>#^^##>#^^##100 cinqturreted-tethrasquare = E100#^^###5 tethrafact How to get it for free? tethrafact sesturreted-tethrasquare = E100#^^###6 septurreted-tethrasquare = E100#^^###7 huiturreted-tethrasquare = E100#^^###8 neufturreted-tethrasquare = E100#^^###9 dexturreted-tethrasquare = E100#^^###10 tethrafact How to get it for free? tethrafact ... Tethracubor Group tethrafact PasteShr tethrafact Now we're getting somewhere. We now introduce the third tetrational operator , ^^###. This operator is referred to by the roots tethratertia, tethracubor, or territertiated... tethracubor = E100#^^###100 (also tethratertia) tethrafact How to get it for free? tethrafact ... once that is established there are plenty of constructions that follow, and some we can obtain with a little bit of tweaking ... here we go again ... grand tethracubor = E100#^^###100#2 tethrafact How to dowload it? tethrafact grangol-carta-tethracubor = E100#^^##100#100 godgahlah-carta-tethracubor = E100#^^##100#^#100 tethrafact How to get it? tethrafact tethrathoth-carta-tethracubor = E100#^^##100#^^#100 tethrasquare-carta-tethracubor = E100#^^###100#^^##100 tethrafact How to get it for free? tethrafact ... duble-tethracubor = E100#^^###100#^^###100 tethrafact How to get it? tethrafact treble-tethracubor = E100#^^###100#^^###100#^^###100 quadruble-tethracubor = E100#^^###*#5 tethrafact How to use it? tethrafact quintuble-tethracubor = E100#^^###*#6 sextuble-tethracubor = E100#^^###*#7 tethrafact How to get it for free? tethrafact septuble-tethracubor = E100#^^###*#8 octuble-tethracubor = E100#^^###*#9 tethrafact PasteShr tethrafact nonuble-tethracubor = E100#^^###*#10 ... tethrafact How to get it for free? tethrafact centuble-tethracubor = E100#^^###*#100 ... tethrafact How to get it for free? tethrafact deutero-tethracubor = E100#^^###*#^^###100 trito-tethracubor = E100#^^###*#^^###*#^^###100 tethrafact How to get it? tethrafact teterto-tethracubor = E100#^^###*#^^###*#^^###*#^^###100 pepto-tethracubor = E100(#^^###)^#5 tethrafact PasteShr tethrafact exto-tethracubor = E100(#^^###)^#6 epto-tethracubor = E100(#^^###)^#7 tethrafact How to get it? tethrafact ogdo-tethracubor = E100(#^^###)^#8 ento-tethracubor = E100(#^^###)^#9 tethrafact How to dowload it? tethrafact dekato-tethracubor = E100(#^^###)^#10 ... tethrafact How to dowload it? tethrafact isosto-tethracubor = E100(#^^###)^#20 ... tethrafact How to dowload it? tethrafact hecato-tethracubor = E100(#^^###)^#100 here are some less immediate ones we can get with a little bit of modification from earlier googolism's ... tethrafact How to use it? tethrafact catenatethracubor = E100(#^^###)^#100 tethracuborgridulus = E100(#^^###)^##100 tethrafact How to get it? tethrafact tethracuborcubiculus = E100(#^^###)^###100 tethracuborquarticulus = E100(#^^###)^####100 tethrafact How to use it? tethrafact tethracuborquinticulus = E100(#^^###)^#^#5 tethracuborsexticulus = E100(#^^###)^#^#6 tethrafact How to get it for free? tethrafact tethracuborsepticulus = E100(#^^###)^#^#7 tethracuborocticulus = E100(#^^###)^#^#8 tethrafact How to get it for free? tethrafact tethracubornoniculus = E100(#^^###)^#^#9 tethracubordeciculus = E100(#^^###)^#^#10 tethrafact How to get it? tethrafact ... tethracuborgodgathored = E100(#^^###)^#^#100 tethrafact PasteShr tethrafact tethracuborgralgathored = E100(#^^###)^#^##100 ... tethrafact How to use it? tethrafact tethracuborgodtotholed = E100(#^^###)^#^#^#100 ... here I'll introduce a new root operator. The virtua-operator takes the expression on the left and raises it to the power of the right (virtua is latin for power)... tethrafact How to use it? tethrafact tethracubor-virtua-tethrathoth = E100(#^^###)^#^^#100 tethracubor-virtua-tethriterator = E100(#^^###)^#^^#>#100 tethrafact PasteShr tethrafact tethracubor-virtua-tethrasquare = E100(#^^###)^#^^##100 tethracubor-virtua-tethrasquared-iterator = E100(#^^###)^#^^##>#100 tethrafact How to dowload it? tethrafact tethracubor-virtua-tethracubor = E100(#^^###)^(#^^###)100 ... we'll also want to be able to tetrate hyperials ... tethrafact How to get it? tethrafact tethracubor-deuterotetrated = E100(#^^###)^(#^^###)100 tethracubor-tritotetrated = E100(#^^###)^(#^^###)^(#^^###)100 tethrafact How to dowload it? tethrafact tethracubor-tetertotetrated = E100(#^^###)^^#4 tethracubor-peptotetrated = E100(#^^###)^^#5 tethrafact How to use it? tethrafact tethracubor-extotetrated = E100(#^^###)^^#6 tethracubor-eptotetrated = E100(#^^###)^^#7 tethrafact How to dowload it? tethrafact tethracubor-ogdotetrated = E100(#^^###)^^#8 tethracubor-entotetrated = E100(#^^###)^^#9 tethrafact How to dowload it? tethrafact tethracubor-dekatotetrated = E100(#^^###)^^#10 ... tethrafact How to get it? tethrafact tethracubor-isostotetrated = E100(#^^###)^^#20 ... tethrafact How to get it for free? tethrafact terrible tethracubor = E100(#^^###)^^#100 terrible terrible tethracubor = E100((#^^###)^^#)^^#100 tethrafact PasteShr tethrafact hundred-ex-terrible tethracubor = E100(#^^###)^^#>#100 ... some further diagonalizations ... tethrafact How to use it? tethrafact tethritertethracubor = E100(#^^###)^^#>#100 duturreted-territethracubor = E100(#^^###)^^#>(#^^###)^^#100 tethrafact How to use it? tethrafact triturreted-territethracubor = E100(#^^###)^^#>(#^^###)^^#>(#^^###)^^#100 tetraturreted-territethracubor = E100(#^^###)^^##4 tethrafact How to get it for free? tethrafact penturreted-territethracubor = E100(#^^###)^^##5 hexturreted-territethracubor = E100(#^^###)^^##6 tethrafact How to get it? tethrafact hepturreted-territethracubor = E100(#^^###)^^##7 ogturreted-territethracubor = E100(#^^###)^^##8 tethrafact How to get it? tethrafact enturreted-territethracubor = E100(#^^###)^^##9 dekturreted-territethracubor = E100(#^^###)^^##10 tethrafact How to dowload it? tethrafact ... isosturreted-territethracubor = E100(#^^###)^^##20 tethrafact How to use it? tethrafact ... hecturreted-territethracubor = E100(#^^###)^^##100 tethrafact PasteShr tethrafact ... on a higher diagonalization ... terrisquared-tethracubor = E100(#^^###)^^##100 tethrafact How to get it? tethrafact terrible terrisquared-tethracubor = E100((#^^###)^^##)^^#100 ... tethrafact How to get it for free? tethrafact territerated terrisquared-tethracubor = E100((#^^###)^^##)^^#>#100 double-terrisquared-tethracubor = E100((#^^###)^^##)^^##100 tethrafact How to dowload it? tethrafact triple-terrisquared-tethracubor = E100(((#^^###)^^##)^^##)^^##100 quadruple-terrisquared-tethracubor = E100(#^^###)^^##>#4 tethrafact How to use it? tethrafact quintuple-terrisquared-tethracubor = E100(#^^###)^^##>#5 sextuple-terrisquared-tethracubor = E100(#^^###)^^##>#6 tethrafact PasteShr tethrafact septuple-terrisquared-tethracubor = E100(#^^###)^^##>#7 octuple-terrisquared-tethracubor = E100(#^^###)^^##>#8 tethrafact How to get it for free? tethrafact nonuple-terrisquared-tethracubor = E100(#^^###)^^##>#9 decuple-terrisquared-tethracubor = E100(#^^###)^^##>#10 tethrafact PasteShr tethrafact ... vigintuple-terrisquared-tethracubor = E100(#^^###)^^##>#20 tethrafact How to get it for free? tethrafact ... centuple-terrisquared-tethracubor = E100(#^^###)^^##>#100 tethrafact How to dowload it? tethrafact ... on a higher diagonalization ... terrisquarediter-tethracubor = E100(#^^###)^^##>#100 tethrafact PasteShr tethrafact ... duturreted-terrisquared-tethracubor = E100(#^^###)^^##>(#^^###)^^##100 tethrafact How to dowload it? tethrafact triturreted-terrisquared-tethracubor = E100(#^^###)^^###3 tetraturreted-terrisquared-tethracubor = E100(#^^###)^^###4 tethrafact How to dowload it? tethrafact penturreted-terrisquared-tethracubor = E100(#^^###)^^###5 hexturreted-terrisquared-tethracubor = E100(#^^###)^^###6 tethrafact How to get it for free? tethrafact hepturreted-terrisquared-tethracubor = E100(#^^###)^^###7 ogturreted-terrisquared-tethracubor = E100(#^^###)^^###8 tethrafact How to get it? tethrafact enturreted-terrisquared-tethracubor = E100(#^^###)^^###9 dekturreted-terrisquared-tethracubor = E100(#^^###)^^###10 tethrafact PasteShr tethrafact ... isosturreted-terrisquared-tethracubor = E100(#^^###)^^###20 tethrafact How to get it for free? tethrafact ... hecturreted-terrisquared-tethracubor = E100(#^^###)^^###100 tethrafact How to get it? tethrafact ... next level ... tethradoplicubor = E100(#^^###)^^###100 tethrafact How to use it? tethrafact terrible tethradoplicubor = E100((#^^###)^^###)^^#100 terrisquared-tethradoplicubor = E100((#^^###)^^###)^^##100 tethrafact How to get it for free? tethrafact ... tethratroplicubor = E100((#^^###)^^###)^^###100 tethrafact How to use it? tethrafact tethraquadroplicubor = E100(((#^^###)^^###)^^###)^^###100 tethraquintoplicubor = E100#^^###>#5 tethrafact PasteShr tethrafact tethrasextoplicubor = E100#^^###>#6 tethraseptoplicubor = E100#^^###>#7 tethrafact How to get it? tethrafact tethra-octoplicubor = E100#^^###>#8 tethranonoplicubor = E100#^^###>#9 tethrafact How to dowload it? tethrafact tethradecoplicubor = E100#^^###>#10 ... tethrafact How to dowload it? tethrafact tethravigintoplicubor = E100#^^###>#20 ... tethrafact How to get it for free? tethrafact tethracentoplicubor = E100#^^###>#100 ... and on to the next level ... tethrafact PasteShr tethrafact tethra-itercubor = E100#^^###>#100 tethra-ditercubor = E100#^^###>(#+#)100 tethrafact How to dowload it? tethrafact tethra-tritercubor = E100#^^###>(#+#+#)100 tethra-quaditercubor = E100#^^###>(#+#+#+#)100 tethrafact PasteShr tethrafact tethra-quiditercubor = E100#^^###>##5 tethra-siditercubor = E100#^^###>##6 tethrafact How to dowload it? tethrafact tethra-sepitercubor = E100#^^###>##7 tethra-ogditercubor = E100#^^###>##8 tethrafact PasteShr tethrafact tethra-nonitercubor = E100#^^###>##9 tethra-decitercubor = E100#^^###>##10 tethrafact How to use it? tethrafact ... tethra-vigintitercubor = E100#^^###>##20 tethrafact How to get it for free? tethrafact ... tethra-centitercubor = E100#^^###>##100 tethrafact How to get it? tethrafact ... on a higher level ... tethra-griditercubor = E100#^^###>##100 tethrafact How to use it? tethrafact tethra-cubiculated-cubor = E100#^^###>###100 tethra-quarticulated-cubor = E100#^^###>####100 tethrafact How to dowload it? tethrafact tethra-quinticulated-cubor = E100#^^###>#^#5 tethra-sexticulated-cubor = E100#^^###>#^#6 tethrafact How to get it? tethrafact tethra-septiculated-cubor = E100#^^###>#^#7 tethra-octiculated-cubor = E100#^^###>#^#8 tethrafact How to get it for free? tethrafact tethra-noniculated-cubor = E100#^^###>#^#9 tethra-deciculated-cubor = E100#^^###>#^#10 tethrafact How to dowload it? tethrafact ... tethra-viginticulated-cubor = E100#^^###>#^#20 tethrafact How to use it? tethrafact ... tethra-centiculated-cubor = E100#^^###>#^#100 tethrafact How to dowload it? tethrafact ... on yet another level ... duturreted-tethracubor = E100#^^###>#^^###100 tethrafact How to dowload it? tethrafact triturreted-tethracubor = E100#^^###>#^^###>#^^###100 tetraturreted-tethracubor = E100#^^###>#^^###>#^^###>#^^###100 tethrafact PasteShr tethrafact penturreted-tethracubor = E100#^^####5 hexturreted-tethracubor = E100#^^####6 tethrafact PasteShr tethrafact hepturreted-tethracubor = E100#^^####7 ogturreted-tethracubor = E100#^^####8 tethrafact How to get it? tethrafact enturreted-tethracubor = E100#^^####9 dekturreted-tethracubor = E100#^^####10 tethrafact How to get it? tethrafact ... isosturreted-tethracubor = E100#^^####20 tethrafact PasteShr tethrafact ... tethrafact How to get it? tethrafact Tethrachoronicor Group A 4-dimensional figure is popularly referred to as a polychoron. The next number can be thought of as employing (in very loose terms mind you) a 4th dimensional power tower of hyperions. Namely it uses the delimiter #^^#4 which is equivalent to the notion of the ordinal w^^w4. From this I derive the came tethrachoronicor ... tethrafact PasteShr tethrafact tethrachoronicor = E100#^^####100 (from tetration + polychoron) tethrafact How to dowload it? tethrafact Of coarse we can derive a whole family of googolism's based on extending tethrachoronicor using all the previously established constructions. Here is just a sampling... grand tethrachoronicor = E100#^^####100#2 tethrafact How to use it? tethrafact ... grangol-carta-tethrachoronicor = E100#^^####100#100 tethrafact How to get it? tethrafact ... tethrachronicor-doplus = E100#^^####100#^^####100 tethrafact How to get it for free? tethrafact tethrachoronicor-triplus = E100#^^####100#^^####100#^^####100 tethrachoronicor-quadruplus = E100#^^####*#5 tethrafact How to get it? tethrafact tethrachoronicor-quintuplus = E100#^^####*#6 tethrachoronicor-sextuplus = E100#^^####*#7 tethrafact How to use it? tethrafact tethrachoronicor-septuplus = E100#^^####*#8 tethrachoronicor-octuplus = E100#^^####*#9 tethrafact How to dowload it? tethrafact tethrachoronicor-nonuplus = E100#^^####*#10 tethrachoronicor-decuplus = E100#^^####*#11 tethrafact How to get it for free? tethrafact ... deutero-tethrachoronicor = E100#^^####*#^^####100 tethrafact How to dowload it? tethrafact trito-tethrachoronicor = E100#^^####*#^^####*#^^####100 teterto-tethrachoronicor = E100(#^^####)^#4 tethrafact PasteShr tethrafact pepto-tethrachoronicor = E100(#^^####)^#5 exto-tethrachoronicor = E100(#^^####)^#6 tethrafact How to get it? tethrafact epto-tethrachoronicor = E100(#^^####)^#7 ogdo-tethrachoronicor = E100(#^^####)^#8 tethrafact How to use it? tethrafact ento-tethrachoronicor = E100(#^^####)^#9 dekato-tethrachoronicor = E100(#^^####)^#10 tethrafact How to get it? tethrafact ... catenatethrachoronicor = E100(#^^####)^#100 tethrafact PasteShr tethrafact tethrachoronigrid = E100(#^^####)^##100 tethrachoronicube = E100(#^^####)^###100 tethrafact How to dowload it? tethrafact tethrachoronitessor = E100(#^^####)^####100 tethrachoronipentor = E100(#^^####)^#^#5 tethrafact How to get it? tethrafact tethrachoronihexor = E100(#^^####)^#^#6 tethrachoroniheptor = E100(#^^####)^#^#7 tethrafact How to use it? tethrafact tethrachoroni-octor = E100(#^^####)^#^#8 tethrachoroni-entor = E100(#^^####)^#^#9 tethrafact PasteShr tethrafact tethrachoroni-dector = E100(#^^####)^#^#10 ... tethrafact PasteShr tethrafact deutetrated-tethrachoronicor = E100(#^^####)^(#^^####)100 tritetrated-tethrachoronicor = E100(#^^####)^(#^^####)^(#^^####)100 tethrafact PasteShr tethrafact tetratetrated-tethrachoronicor = E100(#^^####)^^#4 pentatetrated-tethrachoronicor = E100(#^^####)^^#5 tethrafact How to dowload it? tethrafact hexatetrated-tethrachoronicor = E100(#^^####)^^#6 heptatetrated-tethrachoronicor = E100(#^^####)^^#7 tethrafact How to get it? tethrafact octatetrated-tethrachoronicor = E100(#^^####)^^#8 ennatetrated-tethrachoronicor = E100(#^^####)^^#9 tethrafact How to get it? tethrafact dekatetrated-tethrachoronicor = E100(#^^####)^^#10 ... tethrafact PasteShr tethrafact terrible tethrachoronicor = E100(#^^####)^^#100 terrible terrible tethrachoronicor = E100((#^^####)^^#)^^#100 tethrafact PasteShr tethrafact ... territerated-tethrachoronicor = E100(#^^####)^^#>#100 tethrafact How to get it? tethrafact terriditerated-tethrachoronicor = E100(#^^####)^^#>(#+#)100 territriterated-tethrachoronicor = E100(#^^####)^^#>(#+#+#)100 tethrafact How to get it for free? tethrafact ... duturreted-territethrachoronicor = E100(#^^####)^^#>(#^^####)^^#100 tethrafact How to use it? tethrafact triturreted-territethrachoronicor = E100(#^^####)^^##3 tetraturreted-territethrachoronicor = E100(#^^####)^^##4 tethrafact How to get it? tethrafact ... terrisquared-tethrachoronicor = E100(#^^####)^^##100 tethrafact PasteShr tethrafact terrible terrisquared-tethrachoronicor = E100((#^^####)^^##)^^#100 double terrisquared-tethrachoronicor = E100((#^^####)^^##)^^##100 tethrafact PasteShr tethrafact terrible double terrisquared-tethrachoronicor = E100(((#^^####)^^##)^^##)^^#100 triple terrisquared-tethrachoronicor = E100(#^^####)^^##>#3 tethrafact How to use it? tethrafact quadruple terrisquared-tethrachoronicor = E100(#^^####)^^##>#4 quintuple terrisquared-tethrachoronicor = E100(#^^####)^^##>#5 tethrafact How to get it? tethrafact sextuple terrisquared-tethrachoronicor = E100(#^^####)^^##>#6 septuple terrisquared-tethrachoronicor = E100(#^^####)^^##>#7 tethrafact PasteShr tethrafact octuple terrisquared-tethrachoronicor = E100(#^^####)^^##>#8 nonuple terrisquared-tethrachoronicor = E100(#^^####)^^##>#9 tethrafact How to get it? tethrafact decuple terrisquared-tethrachoronicor = E100(#^^####)^^##>#10 ... tethrafact How to get it for free? tethrafact duturreted-terrisquared-tethrachoronicor = E100(#^^####)^^##>(#^^####)^^##100 triturreted-terrisquared-tethrachoronicor = E100(#^^####)^^###3 tethrafact How to get it? tethrafact tetraturreted-terrisquared-tethrachoronicor = E100(#^^####)^^###4 ... tethrafact How to use it? tethrafact terricubed-tethrachoronicor = E100(#^^####)^^###100 double terricubed-tethrachoronicor = E100((#^^####)^^###)^^###100 tethrafact How to get it? tethrafact triple terricubed-tethrachoronicor = E100(#^^####)^^###>#3 ... tethrafact How to use it? tethrafact duturreted-terricubed-tethrachoronicor = E100(#^^####)^^###>(#^^####)^^###100 triturreted-terricubed-tethrachoronicor = E100(#^^####)^^####3 tethrafact How to use it? tethrafact tetraturreted-terricubed-tethrachoronicor = E100(#^^####)^^####4 ... tethrafact How to dowload it? tethrafact tethra-deuterchoronicor = E100(#^^####)^^####100 tethra-tritochoronicor = E100((#^^####)^^####)^^####100 tethrafact PasteShr tethrafact tethra-tetertochoronicor = E100#^^####>#4 tethra-peptochoronicor = E100#^^####>#5 tethrafact How to get it for free? tethrafact tethra-extochoronicor = E100#^^####>#6 tethra-eptochoronicor = E100#^^####>#7 tethrafact PasteShr tethrafact tethra-ogdochoronicor = E100#^^####>#8 tethra-entochoronicor = E100#^^####>#9 tethrafact PasteShr tethrafact tethra-dekatochoronicor = E100#^^####>#10 ... tethrafact How to use it? tethrafact duturreted-tethrachoronicor = E100#^^####>#^^####100 triturreted-tethrachoronicor = E100#^^####>#^^####>#^^####100 tethrafact How to use it? tethrafact tetraturreted-tetrhachoronicor = E100#^^#####4 ... tethrafact How to dowload it? tethrafact tethrateronicor group ... as you can probably tell we have already settled into a regular pattern. I'll only give a smattering of googolism's possible for the next few groups. Keep in mind the pattern we've just established is still just the very beginning. We haven't even gotten to the pentational cases yet. We've still got a long way to go ... tethrafact How to get it for free? tethrafact tethrateronicor = E100#^^#^#5 tethrafact PasteShr tethrafact (from tetration + polyteron ) note: for rest of examples, resolve fundamental sequences before proceeding tethrafact How to dowload it? tethrafact grand tethrateronicor = E100#^^#5100#2 ... tethrafact How to get it for free? tethrafact grangol-carta-tethrateronicor = E100#^^#5100#100 ... tethrafact PasteShr tethrafact tethrateronicor-doplus = E100#^^#5100#^^#5100 ... tethrafact How to get it? tethrafact deutero-tethrateronicor = E100#^^#5*#^^#5100 ... tethrafact How to dowload it? tethrafact catenatethrateronicor = E100(#^^#5)^#100 ... tethrafact How to dowload it? tethrafact deutetrated-tethrateronicor = E100(#^^#5)^(#^^#5)100 ... tethrafact PasteShr tethrafact terrible tethrateronicor = E100(#^^#5)^^#100 ... tethrafact How to use it? tethrafact territerated tethrateronicor = E100(#^^#5)^^#>#100 ... tethrafact How to use it? tethrafact duturreted-territethrateronicor = E100(#^^#5)^^#>(#^^#5)^^#100 ... tethrafact PasteShr tethrafact terrisquared-tethrateronicor = E100(#^^#5)^^##100 terricubed-tethrateronicor = E100(#^^#5)^^###100 tethrafact PasteShr tethrafact territesserated-tethrateronicor = E100(#^^#5)^^####100 tethra-deuterteronicor = E100(#^^#5)^^#5100 tethrafact How to use it? tethrafact ... tethra-iterteronicor = E100#^^#5>#100 tethrafact PasteShr tethrafact ... duturreted-tethrateronicor = E100#^^#5>#^^#5100 tethrafact How to dowload it? tethrafact triturreted-tethrateronicor = E100#^^#5>#^^#5>#^^#5100 tetraturreted-tethrateronicor = E100#^^#5>#^^#5>#^^#5>#^^#5100 tethrafact PasteShr tethrafact ... tethrapetonicor group tethrafact How to dowload it? tethrafact tethrapetonicor = E100#^^#^#6 (from tetration + polypeton ) tethrafact PasteShr tethrafact grand tethrapetonicor = E100#^^#^(6)100#2 ... tethrafact How to use it? tethrafact grangol-carta-tethrapetonicor = E100#^^#^(6)100#100 ... tethrafact How to dowload it? tethrafact tethrapetonicor-doplus = E100#^^#^(6)100#^^#^(6)100 ... tethrafact How to use it? tethrafact deutero-tethrapetonicor = E100#^^#^(6)*#^^#^(6)100 ... tethrafact PasteShr tethrafact catenatethrapetonicor = E100(#^^#^(6))^#100 ... tethrafact PasteShr tethrafact deutetrated-tethrapetonicor = E100(#^^#^(6))^(#^^#^(6))100 ... tethrafact How to use it? tethrafact terrible tethrapetonicor = E100(#^^#^(6))^^#100 ... tethrafact How to use it? tethrafact territerated tethrapetonicor = E100(#^^#^(6))^^#>#100 ... tethrafact PasteShr tethrafact duturreted-territethrapetonicor = E100(#^^#^(6))^^#>(#^^#^(6))^^#100 ... tethrafact How to dowload it? tethrafact terrisquared-tethrapetonicor = E100(#^^#^(6))^^##100 terricubed-tethrapetonicor = E100(#^^#^(6))^^###100 tethrafact PasteShr tethrafact territesserated-tethrapetonicor = E100(#^^#^(6))^^####100 terripenterated-tethrapetonicor = E100(#^^#^(6))^^#^(5)100 tethrafact How to dowload it? tethrafact tethra-deuterpetonicor = E100(#^^#^(6))^^#^(6)100 ... tethrafact How to get it for free? tethrafact tethra-iterpetonicor = E100#^^#^(6)>#100 ... tethrafact How to get it? tethrafact duturreted-tethrapetonicor = E100#^^#^(6)>#^^#^(6)100 triturreted-tethrapetonicor = E100#^^#^(6)>#^^#^(6)>#^^#^(6)100 tethrafact How to get it? tethrafact tetraturreted-tethrapetonicor = E100#^^#^(6)>#^^#^(6)>#^^#^(6)>#^^#^(6)100 ... tethrafact How to dowload it? tethrafact tethrahexonicor group tethrafact How to get it? tethrafact tethrahexonicor = E100#^^#^#7 (from tetration + polyecton ) tethrafact How to dowload it? tethrafact grand tethrahexonicor = E100#^^#^(7)100#2 ... tethrafact How to dowload it? tethrafact grangol-carta-tethrahexonicor = E100#^^#^(7)100#100 ... tethrafact How to use it? tethrafact tethrahexonicor-doplus = E100#^^#^(7)100#^^#^(7)100 ... tethrafact PasteShr tethrafact deutero-tethrahexonicor = E100#^^#^(7)*#^^#^(7)100 ... tethrafact How to use it? tethrafact tethrahexonichain = E100(#^^#^(7))^#100 ... tethrafact PasteShr tethrafact deutetrated-tethrahexonicor = E100(#^^#^(7))^(#^^#^(7))100 ... tethrafact How to use it? tethrafact terrible tethrahexonicor = E100(#^^#^(7))^^#100 ... tethrafact How to get it for free? tethrafact territerated tethrahexonicor = E100(#^^#^(7))^^#>#100 ... tethrafact How to use it? tethrafact duturreted-territethrahexonicor = E100(#^^#^(7))^^#>(#^^#^(7))^^#100 ... tethrafact How to use it? tethrafact terrisquared-tethrahexonicor = E100(#^^#^(7))^^##100 terricubed-tethrahexonicor = E100(#^^#^(7))^^###100 tethrafact How to get it? tethrafact territesserated-tethrahexonicor = E100(#^^#^(7))^^####100 terripenterated-tethrahexonicor = E100(#^^#^(7))^^#^(5)100 tethrafact How to get it for free? tethrafact terrihexerated-tethrahexonicor = E100(#^^#^(7))^^#^(6)100 tethra-deuterhexonicor = E100(#^^#^(7))^^#^(7)100 tethrafact How to get it? tethrafact ... tethra-iterhexonicor = E100#^^#^(7)>#100 tethrafact How to get it for free? tethrafact ... duturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)100 tethrafact How to use it? tethrafact triturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)100 tetraturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)>#^^#^(7)100 tethrafact How to use it? tethrafact ... tethraheptonicor group tethrafact How to use it? tethrafact tethraheptonicor = E100#^^#^#8 (from tetration + polyzetton ) tethrafact How to dowload it? tethrafact grand tethraheptonicor = E100#^^#^(8)100#2 tethrafact PasteShr tethrafact ... grangol-carta-tethraheptonicor = E100#^^#^(8)100#100 tethrafact How to get it for free? tethrafact ... tethraheptonicor-doplus = E100#^^#^(8)100#^^#^(8)100 tethrafact How to dowload it? tethrafact ... deutero-tethraheptonicor = E100#^^#^(8)*#^^#^(8)100 tethrafact How to use it? tethrafact ... tethraheptonifact = E100(#^^#^(8))^#100 tethrafact How to get it? tethrafact ... deutetrated-tethraheptonicor = E100(#^^#^(8))^(#^^#^(8))100 tethrafact How to dowload it? tethrafact ... terrible tethraheptonicor = E100(#^^#^(8))^^#100 tethrafact How to get it for free? tethrafact ... territerated tethraheptonicor = E100(#^^#^(8))^^#>#100 tethrafact How to use it? tethrafact ... duturreted-territethraheptonicor = E100(#^^#^(8))^^#>(#^^#^(8))^^#100 tethrafact How to get it for free? tethrafact ... terrisquared-tethraheptonicor = E100(#^^#^(8))^^##100 tethrafact How to get it? tethrafact terricubed-tethraheptonicor= E100(#^^#^(8))^^###100 territesserated-tethraheptonicor = E100(#^^#^(8))^^####100 tethrafact PasteShr tethrafact terripenterated-tethraheptonicor = E100(#^^#^(8))^^#^(5)100 terrihexerated-tethraheptonicor = E100(#^^#^(8))^^#^(6)100 tethrafact How to get it? tethrafact terrihepterated-tethraheptonicor = E100(#^^#^(8))^^#^(7)100 tethra-deuterheptonicor = E100(#^^#^(8))^^#^(8)100 tethrafact How to get it? tethrafact ... tethra-iterheptonicor = E100#^^#^(8)>#100 tethrafact How to get it for free? tethrafact ... duturreted-tethraheptonicor = E100#^^#^(8)>#^^#^(8)100 tethrafact How to get it for free? tethrafact triturreted-tethraheptonicor = E100#^^#^(8)>#^^#^(8)>#^^#^(8)100 tetraturreted-tethraheptonicor = E100#^^#^(8)>#^^#^(8)>#^^#^(8)>#^^#^(8)100 tethrafact How to get it for free? tethrafact ... tethra-ogdonicor group tethrafact How to use it? tethrafact ############################################################# ########################################################### tethra-ogdonicor = E100#^^#^#9 tethrahexonicor = E100#^^#^#7 tethrafact How to get it? tethrafact (from tetration + polyecton ) tethrafact How to get it? tethrafact grand tethrahexonicor = E100#^^#^(7)100#2 ... tethrafact How to use it? tethrafact grangol-carta-tethrahexonicor = E100#^^#^(7)100#100 ... tethrafact How to get it for free? tethrafact tethrahexonicor-doplus = E100#^^#^(7)100#^^#^(7)100 ... tethrafact PasteShr tethrafact deutero-tethrahexonicor = E100#^^#^(7)*#^^#^(7)100 ... tethrafact How to get it for free? tethrafact tethrahexonifact = E100(#^^#^(7))^#100 ... tethrafact How to get it for free? tethrafact deutetrated-tethrahexonicor = E100(#^^#^(7))^(#^^#^(7))100 ... tethrafact How to dowload it? tethrafact terrible tethrahexonicor = E100(#^^#^(7))^^#100 ... tethrafact PasteShr tethrafact territerated tethrahexonicor = E100(#^^#^(7))^^#>#100 ... tethrafact How to get it for free? tethrafact duturreted-territethrahexonicor = E100(#^^#^(7))^^#>(#^^#^(7))^^#100 ... tethrafact How to get it? tethrafact terrisquared-tethrahexonicor = E100(#^^#^(7))^^##100 terricubed-tethrahexonicor = E100(#^^#^(7))^^###100 tethrafact PasteShr tethrafact territesserated-tethrahexonicor = E100(#^^#^(7))^^####100 terripenterated-tethrahexonicor = E100(#^^#^(7))^^#^(5)100 tethrafact PasteShr tethrafact terrihexerated-tethrahexonicor = E100(#^^#^(7))^^#^(6)100 tethra-deuterhexonicor = E100(#^^#^(7))^^#^(7)100 tethrafact How to get it? tethrafact ... tethra-iterhexonicor = E100#^^#^(7)>#100 tethrafact How to dowload it? tethrafact ... duturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)100 tethrafact How to get it for free? tethrafact triturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)100 tetraturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)>#^^#^(7)100 tethrafact How to get it? tethrafact ... tethrafact PasteShr tethrafact tethra-ennonicor = E100#^^#^#10 tethrahexonicor = E100#^^#^#7 tethrafact How to dowload it? tethrafact (from tetration + polyecton ) grand tethrahexonicor = E100#^^#^(7)100#2 tethrafact How to dowload it? tethrafact ... grangol-carta-tethrahexonicor = E100#^^#^(7)100#100 tethrafact How to get it for free? tethrafact ... tethrahexonicor-doplus = E100#^^#^(7)100#^^#^(7)100 tethrafact How to dowload it? tethrafact ... deutero-tethrahexonicor = E100#^^#^(7)*#^^#^(7)100 tethrafact How to use it? tethrafact ... tethrahexonifact = E100(#^^#^(7))^#100 tethrafact How to dowload it? tethrafact ... deutetrated-tethrahexonicor = E100(#^^#^(7))^(#^^#^(7))100 tethrafact How to use it? tethrafact ... terrible tethrahexonicor = E100(#^^#^(7))^^#100 tethrafact How to dowload it? tethrafact ... territerated tethrahexonicor = E100(#^^#^(7))^^#>#100 tethrafact How to get it for free? tethrafact ... duturreted-territethrahexonicor = E100(#^^#^(7))^^#>(#^^#^(7))^^#100 tethrafact How to get it? tethrafact ... terrisquared-tethrahexonicor = E100(#^^#^(7))^^##100 tethrafact How to get it? tethrafact terricubed-tethrahexonicor = E100(#^^#^(7))^^###100 territesserated-tethrahexonicor = E100(#^^#^(7))^^####100 tethrafact How to get it for free? tethrafact terripenterated-tethrahexonicor = E100(#^^#^(7))^^#^(5)100 terrihexerated-tethrahexonicor = E100(#^^#^(7))^^#^(6)100 tethrafact How to use it? tethrafact tethra-deuterhexonicor = E100(#^^#^(7))^^#^(7)100 ... tethrafact How to use it? tethrafact tethra-iterhexonicor = E100#^^#^(7)>#100 ... tethrafact PasteShr tethrafact duturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)100 triturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)100 tethrafact How to get it for free? tethrafact tetraturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)>#^^#^(7)100 ... tethrafact How to dowload it? tethrafact tethra-dekonicor = E100#^^#^#11 tethrafact How to use it? tethrafact tethrahexonicor = E100#^^#^#7 (from tetration + polyecton ) tethrafact How to use it? tethrafact grand tethrahexonicor = E100#^^#^(7)100#2 ... tethrafact How to get it for free? tethrafact grangol-carta-tethrahexonicor = E100#^^#^(7)100#100 ... tethrafact How to get it for free? tethrafact tethrahexonicor-doplus = E100#^^#^(7)100#^^#^(7)100 ... tethrafact How to dowload it? tethrafact deutero-tethrahexonicor = E100#^^#^(7)*#^^#^(7)100 ... tethrafact How to get it? tethrafact tethrahexonifact = E100(#^^#^(7))^#100 ... tethrafact PasteShr tethrafact deutetrated-tethrahexonicor = E100(#^^#^(7))^(#^^#^(7))100 ... tethrafact How to get it for free? tethrafact terrible tethrahexonicor = E100(#^^#^(7))^^#100 ... tethrafact How to dowload it? tethrafact territerated tethrahexonicor = E100(#^^#^(7))^^#>#100 ... tethrafact How to use it? tethrafact duturreted-territethrahexonicor = E100(#^^#^(7))^^#>(#^^#^(7))^^#100 ... tethrafact PasteShr tethrafact terrisquared-tethrahexonicor = E100(#^^#^(7))^^##100 terricubed-tethrahexonicor = E100(#^^#^(7))^^###100 tethrafact How to dowload it? tethrafact territesserated-tethrahexonicor = E100(#^^#^(7))^^####100 terripenterated-tethrahexonicor = E100(#^^#^(7))^^#^(5)100 tethrafact How to use it? tethrafact terrihexerated-tethrahexonicor = E100(#^^#^(7))^^#^(6)100 tethra-deuterhexonicor = E100(#^^#^(7))^^#^(7)100 tethrafact PasteShr tethrafact ... tethra-iterhexonicor = E100#^^#^(7)>#100 tethrafact How to use it? tethrafact ... duturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)100 tethrafact PasteShr tethrafact triturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)100 tetraturreted-tethrahexonicor = E100#^^#^(7)>#^^#^(7)>#^^#^(7)>#^^#^(7)100 tethrafact How to get it for free? tethrafact ... tethrafact PasteShr tethrafact tethraspace group tethrahectonicor = E100#^^#^#101 tethrafact How to dowload it? tethrafact tethrachilonicor = E100#^^#^#1001 tethramyrionicor = E100#^^#^#10,001 tethrafact How to get it? tethrafact numbers in need of names E100(#^^#^#)^^#^#100 tethrafact How to get it? tethrafact E100((#^^#^#)^^#^#)^^#^#100 E100#^^(#^#)>#100 tethrafact How to get it for free? tethrafact E100#^^(#^#)>#^^#^#100 E100#^^(#^#*#)100 tethrafact How to get it? tethrafact E100#^^(#^#*##)100 E100#^^(#^#*#^#)100 tethrafact How to use it? tethrafact E100#^^#^##100 E100#^^#^###100 tethrafact PasteShr tethrafact E100#^^#^#^#100 E100#^^#^#^#^#100 tethrafact How to get it for free? tethrafact tethrarxitri = E100#^^#^^#100 (also tethrarxitri from “arx” latin for tower) tethrarxiquad = E100#^^#^^#^^#100 tethrarxiquid = E100#^^^#5 tethrarxissid = E100#^^^#6 tethrafact How to use it? tethrafact tethrarxissip= E100#^^^#7 tethrarxi-ogd = E100#^^^#8 tethrarxideck = E100#^^^#10 pentacularex = E100#^^^#100 (also tethraxihecate) pentacthulum = E100#^^^#100 tethrafact How to get it? tethrafact pentacthulusquare = E100#^^^##100 pentacthulucubor = E100#^^^###100 pentacthulutesseractus = E100#^^^####100 pentacthulupenteractus = E100#^^^#^#5 pentacthuluhexeractus = E100#^^^#^#6 tethrafact PasteShr tethrafact ... pentarxitri = E100#^^^#^^^#100 hexhaxulum = E100#^^^^#100 hexhaxulusquare = E100#^^^^##100 hexhaxulucubor = E100#^^^^###100 tethrafact How to use it? tethrafact hexarxitri = E100#^^^^#^^^^#100 ... tethrarxichill = E100#^^^#1000 tethrarximyr = E100#^^^#10,000 tethrarxi-octad = E100#^^^#100,000,000 tethrafact How to use it? tethrafact tethrarxi-sedeniad = E100#^^^#10,000,000,000,000,000 grand pentacularex = E100#^^^#100#2 = E100#^^^#pentacularex double grand pentacularex = E100#^^^#100#3 triple grand pentacularex = E100#^^^#100#4 pentacula-expoundum = E100#^^^#^#100 tethrafact How to use it? tethrafact pentacula-hyperia-expoundum = E100#^^^#^#^#100 pentacula-duhyperia-expoundum = E100#^^^#^#^#^#100 pentacula-tethrulex = E100#^^^#^^#100 grand pentacula-tethrulex = E100#^^^#^^#pentacula-tethrulex joe pellinger = E100#^^^#^^#^#199,999,999 tethrafact How to use it? tethrafact pentaxitri = E100#^^^#^^^#100 pentaxiquad = E100#^^^#^^^#^^^#100 pentaxiquid = E100#^^^^#5 pentaxissid= E100#^^^^#6 pentaxissip = E100#^^^^#7 tethrafact How to use it? tethrafact pentaxi-ogd = E100#^^^^#8 hexhaxularex = E100#^^^^#100 heptacularex = E100#^^^^^#100 octacularex = E100#^^^^^^#100 goliath = E100#{#}#10 = E100#^^^^^^^^^^#100 tethrafact PasteShr tethrafact godsgodgulus = E100#{#}#100 = E100#^^^…^^^#100 w/100 ^s colossus = E100#{#}#50,000,000,000,000,000 grand godsgodgulus = E100#{#}#100#2 = E100#{#}#godsgodgulus blasphemorgulog = E100{#,#,1,2}100 grand blasphemorgulog = E100{#,#,1,2}100#2 tethrafact How to dowload it? tethrafact sprach Zarathustra = E100{#,#(1)2}50,000,000,000,000,000 Thus sprach Zarathustra = E100{#,#(1)2}50,000,000,000,000,000#2 Zarathustra-Grid = E100{#,#(2)2}50,000,000,000,000,000 Zarathustra-Cube = E100{#,#(3)2}50,000,000,000,000,000 Zarathustra-Tess = E100{#,#(4)2}50,000,000,000,000,000 tethrafact How to use it? tethrafact tethrafact How to get it? tethrafact ######################################################## ######################################################## ######################################################## ######################################################## tethrafact How to dowload it? tethrafact tethriter-tetraplux = E100#^^#>#^^#100 (also tethri-staculardeus) tethri-staculartruce = E100#^^#>#^^#>#^^#100 tethri-stacularquad = E100#^^#>#^^#>#^^#>#^^#100 tethri-stacularquid = E100#^^##5 tethricross = E100#^^##100 tethrafact How to use it? tethrafact grangol-carta-tethricross = E100#^^##100#100 godgahlah-carta-tethricross = E100#^^##100#^#100 tethrathothi-carta-tethricross = E100#^^##100#^^#100 tethriterator-carta-tethricross = E100#^^##100#^^#>#100 tristo-tethricross = E100#^^##100#^^##100 tethrafact How to dowload it? tethrafact deutero-tethricross = E100#^^##*#^^##100 trito-tethricross = E100#^^##*#^^##*#^^##100 terri-tethricross = E100(#^^##)^^#100 territer-tethricross = E100(#^^##)^^#>#100 terriditer-tethricross = E100(#^^##)^^#>(#+#)100 tethrafact How to get it for free? tethrafact tethricube = E100#^^###100 (also tethrurricybus from “turri cybus” latin for “tower cube”) tethritess = E100#^^####100 tethripent = E100#^^#^#5 tethrihex = E100#^^#^#6 tethrihept = E100#^^#^#7 tethrafact How to get it? tethrafact tethri-ocht = E100#^^#^#8 tethrihecate = E100#^^#^#100 tethrichill = E100#^^#^#1000 tethrimyr = E100#^^#^#10,000 tethri-octad = E100#^^#^#100,000,000 tethrafact How to dowload it? tethrafact tethri-sedeniad = E100#^^#^#10,000,000,000,000,000 tethrarxitri = E100#^^#^^#100 (also tethrarxitri from “arx” latin for tower) tethrarxiquad = E100#^^#^^#^^#100 tethrarxiquid = E100#^^^#5 tethrarxissid = E100#^^^#6 tethrafact How to get it? tethrafact tethrarxissip= E100#^^^#7 tethrarxi-ogd = E100#^^^#8 tethrarxideck = E100#^^^#10 pentacularex = E100#^^^#100 (also tethraxihecate) pentacthulum = E100#^^^#100 tethrafact How to dowload it? tethrafact pentacthulusquare = E100#^^^##100 pentacthulucubor = E100#^^^###100 pentacthulutesseractus = E100#^^^####100 pentacthulupenteractus = E100#^^^#^#5 pentacthuluhexeractus = E100#^^^#^#6 tethrafact PasteShr tethrafact ... pentarxitri = E100#^^^#^^^#100 hexhaxulum = E100#^^^^#100 hexhaxulusquare = E100#^^^^##100 hexhaxulucubor = E100#^^^^###100 tethrafact How to get it for free? tethrafact hexarxitri = E100#^^^^#^^^^#100 ... tethrarxichill = E100#^^^#1000 tethrarximyr = E100#^^^#10,000 tethrarxi-octad = E100#^^^#100,000,000 tethrafact How to dowload it? tethrafact tethrarxi-sedeniad = E100#^^^#10,000,000,000,000,000 grand pentacularex = E100#^^^#100#2 = E100#^^^#pentacularex double grand pentacularex = E100#^^^#100#3 triple grand pentacularex = E100#^^^#100#4 pentacula-expoundum = E100#^^^#^#100 tethrafact How to get it? tethrafact pentacula-hyperia-expoundum = E100#^^^#^#^#100 pentacula-duhyperia-expoundum = E100#^^^#^#^#^#100 pentacula-tethrulex = E100#^^^#^^#100 grand pentacula-tethrulex = E100#^^^#^^#pentacula-tethrulex joe pellinger = E100#^^^#^^#^#199,999,999 tethrafact How to dowload it? tethrafact pentaxitri = E100#^^^#^^^#100 pentaxiquad = E100#^^^#^^^#^^^#100 pentaxiquid = E100#^^^^#5 pentaxissid= E100#^^^^#6 pentaxissip = E100#^^^^#7 tethrafact How to dowload it? tethrafact pentaxi-ogd = E100#^^^^#8 hexhaxularex = E100#^^^^#100 heptacularex = E100#^^^^^#100 octacularex = E100#^^^^^^#100 goliath = E100#{#}#10 = E100#^^^^^^^^^^#100 tethrafact PasteShr tethrafact godsgodgulus = E100#{#}#100 = E100#^^^…^^^#100 w/100 ^s colossus = E100#{#}#50,000,000,000,000,000 grand godsgodgulus = E100#{#}#100#2 = E100#{#}#godsgodgulus blasphemorgulog = E100{#,#,1,2}100 grand blasphemorgulog = E100{#,#,1,2}100#2 tethrafact How to get it? tethrafact sprach Zarathustra = E100{#,#(1)2}50,000,000,000,000,000 Thus sprach Zarathustra = E100{#,#(1)2}50,000,000,000,000,000#2 Zarathustra-Grid = E100{#,#(2)2}50,000,000,000,000,000 Zarathustra-Cube = E100{#,#(3)2}50,000,000,000,000,000 Zarathustra-Tess = E100{#,#(4)2}50,000,000,000,000,000 tethrafact How to get it for free? tethrafact ips – greek for “heighten” arx – latin for “castle” turri – latin for “tower” laniatus – latin for “excellent” excellens – excellence (latin) eximus – extraordinary (latin) tethrafact How to use it? tethrafact tethrafact PasteShr tethrafact pentacular = E100#^^^#100 hexhaxular = E100#^^^^#100 tethrafact How to use it? tethrafact heptacular = E100#^^^^^#100 octacular = E100#^^^^^^#100 godsgodgulus = E100#^^^^^^^^^^......^^^^^^^^^^#100 w/100 ^s tethrafact How to use it? tethrafact E100#{#}#100 = E100#{100}#100 (#{#}#)^^# tethrafact How to dowload it? tethrafact (#{#}#)^^^# (#{#}#){#}# #{#}#># tethrafact How to get it for free? tethrafact #{#}## #{#}### tethrafact How to get it? tethrafact #{#}#^# #{#}#{#}#{#}#{#}# #{#}^# tethrafact How to get it? tethrafact #{#}^^# #{#}^^^# tethrafact How to get it? tethrafact #{#}^^^^# #{#+#}# #{#+#+#}# tethrafact How to dowload it? tethrafact #{##}# #{###}# tethrafact PasteShr tethrafact #{#^#}# #{#^^#}# #{#^^^#}# tethrafact PasteShr tethrafact #{#{#}#}# ... tethrafact PasteShr tethrafact blasphemorgulog = E100{#,#,1,2}100 grand blasphemorgulog = E100{#,#,1,2}100#2 tethrafact How to get it? tethrafact tethrafact PasteShr tethrafact tethrafact How to get it for free? tethrafact tethrafact How to use it? tethrafact tethrafact How to dowload it? tethrafact αβγδεζηθικλμνξοπρςστυφχψω αβγδεζηθικλμνξοπρςστυφχψω tethrafact How to use it? tethrafact gulog (a logarithm deals with orders of magnitude a gulogarithm deals with a number of googological magnitude!) godsgodgulog blasphemorgulog tethrafact How to dowload it? tethrafact gulus - for numbers of the googol family of size greater than or equal to order type w^w^w joe pellinger tethrafact How to dowload it? tethrafact Centurion / Grand Centurion / Super Centurion Colossus tethrafact PasteShr tethrafact Colossulus Goliath tethrafact How to use it? tethrafact Og (A name of a giant) Nephilim tethrafact PasteShr tethrafact Nephilus-Megalosus Brobdingnagulus tethrafact How to get it for free? tethrafact Gargantua Pantagruel tethrafact How to dowload it? tethrafact Tremendagulus Godsgodgulus tethrafact How to get it for free? tethrafact blasmephorgulus gralgathor tethrafact How to dowload it? tethrafact tethrathor tetritathor tethrafact How to get it? tethrafact teritathor iteratathor tethrafact How to get it? tethrafact tetration + iterator --> tethriterator = E100#^^#>#100 (alternative name for tethrathoth ba'al) tethrafact How to get it for free? tethrafact grand tethriterator = E100#^^#>#100#2 (alt. of Great and Terrible tethrathoth) tethrafact