User blog comment:Vel!/Yudkowsky on googology/@comment-205.201.126.113-20140402150842/@comment-5982810-20140402173403

@65.26.80.144

sigh ... your confusing 2 completely different notations. My use of ' & ' has nothing to do with Bowers' use of ' & '.

In my case the ampersand-operator is a function mapping ordinals-to-ordinals (specifically it enumerates the fixed points of b --> {b,#+1,1,2} ). This ampserand may be combined with 'stars' to create higher and higher ordinal-to-ordinal functions enumerating the fix points of operators formed with one less star.

Bowers' '&' is known as the array-of-operator. It is not yet fully understood, but it is radically different than other ordinal notations.

These notations mean completely different things and also have completely different signatures. My ampersand-operator takes a single ordinal argument. So expressions like ...


 * ...***& don't mean anything unless you specify an argument ie. ***...***&(1) for example (1 is the smallest argument allowed).

Bowers' array-of-operator however is a binary function in which the first argument can itself be an expression involving the array-of-operator. For example 3&3&3 is understood as (3&3)&3. 3&3 does not evaluate to a numeric value, but rather acts as an ordinal describing an array-space which will get filled by 3's.

Just because Bowers' and I use the ampersand doesn't mean they have anything to do with each other.

The ampersand-operator with 100 stars behind it isn't going to escape the subsystem it's part of. Any 100 star operator will easily be beaten by a 101 star operator. For example...


 * (a)&(b) = ***...***&(b) w/a *s for some ordinal b

So we could roughly interpret your numbers as...

E100*(100)&(*(100)&(*(100)&( ... *(100)&(*(100)&(*(100)&(#))) ... )))100 w/100 &'s

this can be simplified to...

E100#*(101)^#100

Then the rest of your numbers can be defined as...

E100#*(101)^#100#2

E100#*(101)^#100#3

etc.

So these numbers would be vastly smaller than a ''transmorgrifihgh. In fact they would be much less than a grand gorgonghoulgog''.

Now I know what you were trying to do. You wanted to try and use my trick to extend Bowers' array-of-operator. However there are two important things you are overlooking (1) You can not assume that there is a clean translation from my notation to Bowers'. As I said the signatures are different, so it's not entirely clear what a *&-array-of-operator would mean, let alone adding more stars. YOU have to define how this works. You can't just take unrelated ideas and assume that they mesh. You have to actually prove that they do, make any necessary corrections if they don't, and actually describe how this works to others. Don't just assume it's obvious, or that it's a given.

To test that your idea actually makes sense ask yourself this question: What does #*&# expand to? If you can't answer that question then something is seriously wrong and you have to figure out what before you proceed. Googology is a lot like architecture. You can't start building your roof garden if you don't even have a firm foundation to build it on.

(2) Even if you did carry out such a definition ... it wouldn't be a significant improvement on Legion-Space. Think about it, the star notation was used to build a number like gorgonghoulgog and ''transmorgrifihgh. Though vast in their own right, these are vanishingly tiny compare to Something in the vicinity of Sprach Zarathustra. ''At best you would be doing something like adding the Ackermann-ordinal to the ordinal L (the ordinal of Legion-Space). That would barely amount to anything and would be a trivial extension by definition (since the ordinal you're adding is already a member of L). Before we can extend L in a non-trivial way we first have to understand L.

(Sbiis.ExE)