User:Cookiefonster/BEAF

This is my informal proposal on how to work with BEAF past tetrational arrays - since there's a lot of talk about BEAF past epsilon-zero being fully defined, I figured I should jump in with a variant of my own designed to be as unproblematic as possible, with as little deviation from what people do on the wiki as possible.

This page was copied from a blog post of mine, and expanded to a proper user subpage so that it would be recognized as more of a designation than just a random idea. To comment on this proposal, just go the page's talk page. Note that this is an informal proposal - feel free to suggest what to improve on though and how it can be more formalized. It is designed mainly to be as unproblematic as possible. Feel free to criticize (but keep in mind that I'm really terrible at formalizing stuff) or make a formal version of this proposal.

Epsilon-zero to the SVO
Arrays up to tetrational arrays are defined as agreed upon.

Then, there is a new separator, (X^^X), which works as follows:

{a,b(X^^X)n@} = {a,a(#)n@} where (#) is the bth member of the following sequence:

(1), (0,1), ((1)1), ((0,1)1), (((1)1)1), etc.

Then, we define (X^^X*X) to be a new separator:

{a,b(X^^X*X)n@} = {a,a(X^^X)(X^^X)...(X^^X)(X^^X)n-1@} with b copies of (X^^X) - separator mechanics work as agreed upon

Then:

{a,b(X^^X*X^2)n@} = {a,a(X^^X*X)(X^^X*X)...(X^^X*X)(X^^X*X)n-1@} with b copies of (X^^X*X)

The sequence continues with (X^^X*X^3), (X^^X*X^4), etc, and then {a,b(X^^X*X^X)n@} = {a,a(X^^X*X^b)n@}

Continue with {a,b(X^^X*X^(X+1))n@} = {a,a(X^^X*X^X)(X^^X*X^X)...(X^^X*X^X)(X^^X*X^X)n-1@} with b copies of (X^^X*X^X)

and that idea (note that X in these separators behaves exactly like omega in the FGH) can logically take us up to (X^^X*X^^X) = ((X^^X)^2)

Continue starting with ((X^^X)^2*X) as the "next" separator after ((X^^X)^2) just as (X^^X*X) is next after (X^^X), and that takes us to ((X^^X)^2*X^^X) = ((X^^X)^3)

This idea can make sequences clear, such as {a.b((X^^X)^X^X)n@} = {a.a((X^^X)^X^b)n-1@}, etc, until we get to ((X^^X)^(X^^X)^(X^^X).,....^(X^^X)). Then we define {a,b(X^^2X)n@} to be {a,a(#)n@} where # is the bth member of the sequence:

(X^^X), (X^^X)^(X^^X), (X^^X)^(X^^X)^(X^^X), etc

Why did I make that decision? Because n^^2n approximates (n^^n)^^n, and therefore making X^^2X = (X^^X)^^X is pretty unproblematic. There are probably better solutions, but I'll use this one for simplicity's sake.

We can do the same thing with X^^3X = bth member of [(X^^2X), (X^^2X)^(X^^2X), (X^^2X)^(X^^2X)^(X^^2X), etc], etc, until we get X^^^X = bth member of [X, X^^X, X^^X^^X. etc]

Then we can logically continue with similar ideas to get X^^^2X = bth member of [X^^^X, (X^^^X)^^(X^^^X), (X^^^X)^^(X^^^X)^^(X^^X), etc], until we get X^^^^X = bth member of [X, X^^^X, X^^^X^^^X, etc]

We can logically then switch to array notation within separators, e.g. X^^X = {X,X,2}, X^^^X = {X,X,3}, and similar stuff until we get {X,X,X}, which isn't problematic at all - since X behaves like omega we can logically go up to things like

{X,X,X+1} = bth member of [X, {X,X,X}, {X,{X,X,X},X}, {X,{X,{X,X,X},X},X}, etc], and with that sort of stuff we can get:

{X,X,X+2}, {X,X,2X}, {X,X,X^2}, {X,X,X^X}, {X,X,X^^X}, {X,X,{X,X,X}}, etc

The limit of all this is {X,X,1,2}, bth member of the sequence [X, {X,X,X}, {X,X,{X,X,X}}, {X,X,{X,X,{X,X,X}}}, etc].

This is a pretty informal proposal, but it should provide working definitions for the following numbers (after all, notations like /xE^ are informal but still well defined):

Triakulus = {3,3(X^^^X)2} - that solves to:

{3,3(X^^X^^X)2}

= {3,3(X^^X^X^X)2}

= {3,3(X^^X^X^3)2}

= {3,3(X^^X^(X^2*3))2}

= {3,3(X^^X^(X^2*2+X^2*2+X^2*2))2}

etc

Kungulus = {10,100(X^^^X)2}

which solves to

{10,10(X^^X^^X^^.....(100 X's).....X^^X^^X)2}

Quadrunculus = {10,100(X^^^^X)2}

Tridecatrix = {10,10({X.X,10})2}

Humongulus = {10,10({X,X,100}2}

We can continue with {X,2X,1,2} = {{X,X,1,2},X,1,2}, and in general {X,2X,@} = {{X,X,@},X,@} - with that approach we can go all the way up to {X,X(1)2} arrays, equivanent to SVO in the FGH.

From SVO to LVO
Coming soon. Will be based on Hyp cos's BEAF "analysis", and based on the idea that a legion is θ(Ω_ω), not the LVO.