(actually up to SVO now) 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.

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


{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

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^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}


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}

It can't define a golapulus and beyond, but it's at least more wide-ranged than tetrational arrays.

Suggest opinion, how to improve, etc in comments if you want to.

Edit: Actually, 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.

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