User blog comment:Vel!/Yudkowsky on googology/@comment-65.26.80.144-20140401222126/@comment-24.103.234.74-20140401230429

@65.26.80.144

Ummm... assuming that even makes any sense, adding the array-braces doesn't add anything to the expression. {b} = b. The only exception to this is the array-of-operator, which acts as a short hand for arrays. ie. {3&3} = {3,3,3} = 3^^^3. Therefore your numbers can be shortened to...

E100#*^^#100

E100#*^^#100#2 = E100#*^^#(E100#*^^#100)

E100#*^^#100#3 = E100#*^^#(E100#*^^#(E100#*^^#100))

These numbers are NOWHERE NEAR Sprach Zarathustra. They are infintesimally tiny in comparison, so calling them "super", which means above and beyond, doesn't make much sense. Some better names?

astralthrathoth = E100#*^^#100

grand astralthrathoth = E100#*^^#100#2

grand grand astralthrathoth = E100#*^^#100#3

Also I believe your making a classic mistake. It's the same mistake that causes people to invent salad numbers. It may seem that the longer an expression is ... the larger the number. For example a longer decimal is greater than a smaller one. However these are special cases. In general, length in and of itself does not confer superior power. For example the expression 1+1+1+ ... +1+1+1 w/100,000,000 1's is a very very long expression which requires exactly 199,999,999 characters. Yet it is much much smaller than ' E99 ' which only requires 3 characters. Basically numbers contain information. The naive assumption is that the amount of information a number contains is directly proportional to its size. This assumption turns out to be false because repetitive information can be condensed. This is the basis of file compression. Numbers which contain high degrees of self-similar structures and order can be compressed to unimaginable scales. However, for every such highly compressible number, there are neighboring members which can not be compressed any better than logarithmic-space ( in other-words, there is no better notation for them other than to write out their full decimal expansion). These principles are the very reason googology is possible. If the information in numbers was directly proportional to their size, then googology could never venture beyond numbers with exponential numbers of digits (a googolplex would be a good upper-limit in this case). Instead what we discover is that we can compress and compress until numbers so massive they literally dwarf everything known to man, can be fit into the space of a few characters, and defined with only a few pages of text. Keep in mind though that the numbers of googology are the exceptions, not the rule. The vast majority of integers between 0 and any sufficiently large googolism can never be expressed with fewer than say, a googolplex characters. As large as the numbers we can imagine  ... there is necessarily that many numbers whose form is so irregular that they can not be expressed in any fewer characters than the number itself! That is the true terror of googology.

Now this isn't just theoretical ... it has a lot of bearing on how we actually do googology in practice. In general, never repeat character sequences, because they can always be compacted using another notation. The only reason I use the w/-operator extensively on my site is because it conveys to my readers the ''sheer SIZE of the numbers involved. ''People have a hard time visualizing large numbers, and for the numbers we deal with that is literally impossible. But if you can imagine just how long the expression becomes, and then realize that every step in that expression involves unfathomable jumps in the numbers ... o_0; ... well let's just say you can feel like your head is going to explode. In contrast, googology makes it possible to write compactly such numbers in very short expressions like ' E100#*^^#100 ' which only requires 12 characters. Written that way it doesn't look nearly as impressive ... but start unpacking it and you begin to realize the truly terrifying power that is locked away in that little pandora's box!

In short, you never have to use the w/-operator and it is in fact no better than basic-iteration! It is only used as a place holder notation until better methods are developed, or to illustrate how expressions will expand.

(Sbiis.ExE)