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If you want to go past ε_0, you can make a notation that includes ε_x, or for more power, Veblen functions or ordinal collapsing functions (which my f uses). I already posted some information regarding a method of creating an egç in another thread.

In the surreal numbers, ω-1 is well defined, is greater than all finite numbers, and is less than ω.

(a,b)= (ω^a)+b, ω^a >= b b+(ω^a), ω^a < b ()= 0 Would be one way to do it, in a function notation. You only need one of the pair cases to represent all ordinals, and using b+(ω^a) makes FS functions simpler. How does one turn the fgh level into a function without using the fgh itself? You can also ...

trees :D So how would I use that to make a vgç function? Or would it be better just to define a new, simpler function? A good start would be to make a function for computing fundamental sequences in this tree representation. Also, if possible, simplifying functions is usually a good idea. [quote="M...

One possible method of encoding ordinals as binary trees is to use a leaf node to represent 0 and a branch node to represent b+w^a, where a and b are its children. For example, 1=0+w^0 would be (()()) using a linear representation of trees. 2=1+w^0=(()(()())), w=0+w^1=((()())()), w+1=w+w^0=(((()())(...

One thing that may be useful is that p(a,b)=2^a*(2b+1) is different for all (nonnegative integer) values of a and b, so you can encode two numbers into one. Also, it never outputs 0, so you can do something different in that case. Using that, you can encode arbitrary binary trees into nonnegative in...

There seem to be 2 primitive types of puffers in the patterns in that post, which is the same as the number of single switch engine stabilizations in Life.

Well, you can do something like my f function, where you encode FGH levels as numbers, or do stuff like VECTORPARTY or TREE. I don't think there's any other way to get high FGH levels without doing one of those or using array notation methods like in BEAF.

You either need infinitely many arguments (which will lead you down the array notation path) or more complex methods.

Speaking of times when Gustavo did something possibly useful, what's the most useful thing Gustavo has ever done?

SSCG(3) with 2 node colors and 1--2 forbidden is at least SSCG(SSCG(3)) since you can construct SSCG(3) graphs using only color 1 and SSCG(SSCG(3)) graphs using only color 2, which is far larger than SSCG(4). (Also, SSCG(3)>TREE(3) as demonstrated by APG in a cp4space post) My best sequence for LSSC...

It's probably at most w^3.

True, the 1st is something along the lines of 2^^^^^^^^^^^^^^^^^^^^^^^^^n*n!^^^^^...^^^n!, but the 2nd is much, much larger. I think I actually disagree with fluffykitty on this; you can experiment with 4 functions instead of 26. Actually, S∈P(S) so all of the functions in that post are infinite. I...

I don't think anyone has thought of that before and I'm not sure if it's finite. If it is, it's probably much stronger than SSCG(n). One method to demonstrate FGH lower bounds for this type of function is to create a transfinite list of graphs/trees/vectors/whatever such that no element is contained...

If you're deleting this thread, why not delete GOL-SSRP or "What is sesame oil?"?

2. q(a)=FGHordinal(b=>f(a,b)) using my f function

TREE(TREE(TREE(…n))),where "n" is also the number of instances of "TREE(," is MUCH larger than TREE(n) for all positive integers n that are greater than or equal to 2. Also, I don’t think that’s much higher in the hierarchy than TREE(n), as, according to my extremely limited understanding of what f...

"logical ENGLISH sentence"

I think you need to be significantly more precise with your definitions to produce a concrete function.

The quick brown fox jumped over the lazy dog

H(X) = 4.36852

Defining 5 functions per w2 FGH levels (I missed the fact that ç also adds w levels) is unnecessarily complex. Instead, you can use f, ç but with ltr_x(x,x,x) replaced with f(x,x,x), elç but with elltr replaced with the modified ç, etc. Doing that would make it much simpler to create a w^2 level fun...

Also, here's a second p58 wick component: x = 57, y = 19, rule = B3-cqy4j/S2-n34i 2o53b2o$o14b3o25b3o10bo$bob2o10bobo25bobo6b2obo$2obo11bobo18b2o5bobo7b ob2o$bob2o28b2o17b2obo$bobo5bo23bo19bobo$2ob2o5bo4b2o26b2o7b2ob2o$9bo 4bobo4bo14b3o3bobo$14bobo3bo11b2o2bobo3bobo$24bo7b2o4b2o$18b2o4bo$6bo 2b2o8bo...

Do something similar to going from f,g... to ltr_n but with ltr, elltr, rlltr, ulltr, plltr, tlltr... (since apparently vlltr does not exist)

For that you need a single definition for all levels of f, ltr, a definition for all levels of ç, a definition for all levels of Mltrs, and a definition for all levels of σM which call each other recursively. It might simplify things to remove some functions of each level (since you can just add mor...