Binary Big Number Systems

[0] 1 11 [2] 111 [3] 1111 [4]

1.#n. = n P; = P [= P;0]

1a) a;1 = aa [a+a] 1b) = a.#a. [a*a] a;11 = (a;1);1 [1a) a*4] 2a) a;111 = (a;1);11 [1a) a*2^3] 2b) = (a;11);11 [1b) a^(2^4)] 2a) a;b1 = (a;1);b [a;b =1a) a*2^b =1b) a^(2^b)] 2b) = (a;b);b [=1a) a*2^(2^b) =1b) a^(2^(2^b))] 2a1) a;b;c1 = a;(b;1);c 2a2) = (a;1);(b;1);c 2b2) = (a;b;c);(a;b;c);c 2a1) a;...;m;n1 = a;...;(m;1);n 2a2) = (a;1);...;(m;1);n A=(a;...;m;n) 2b2) = A;...;A;n 1a) A;;1 = A;A 1b) = A;...;A [A#A] 2a) A;;b1 = (A;;1);;b 2b) = (A;;b);;b 2a1) A;;...;;M;;z1 = A;;...;;(M;;1);;z 2a2) = (A;;1);;...;;(M;;1);;z P=(A;;...;;M;;z) 2b2) = P;;...;;P;;z

1a) A;.#r1.1 = A;.#r.A 1b) = A;.#r....;.#r.A [A#A]

2a) A;.#r.b1 = (A;.#r.1);.#r.b 2b) = (A;.#r.b);.#r.b

2a1) A;.#r....;.#r.M;.#r.z1 = A;.#r....;.#r.(M;.#r.1);.#r.z 2a2) = (A;.#r.1);.#r....;.#r.(M;.#r.1);.#r.z P=(A;.#r....;.#r.M;.#r.z) 2b2) = P;.#r....;.#r.P;.#r.z

;a = a ;a;b = ab [a+b]

;a;b;c = a;.#c.b

(1a ;a;1;1;1 = a;_1,1 = a;.#a.a = ;a;a;a ;a;11;1;1 = a;_1,11 = (a;_1,1);_1,1 = ;(;a;a;a);(;a;a;a);(;a;a;a) 2a) ;a;b1;1;1 = a;_1,b1 = (a;_1,1);_1,b = ;(;a;1;1;1);b;1;1 2b) = (a;_1,b);_1,b = ;(;a;b;1;1);b;1;1 ;a;1;11;1 = a;_1,;_1,1 = a;_1,a = ;a;a;1;1 2a) ;a;b1;11;1 = a;_1,;_1,b1 = (a;_1,;_1,1);_1,;_1,b = ;(;a;1;11;1);b;11;1 2b) = (a;_1,;_1,b);_1,;_1,b = ;(;a;b;11;1);b;11;1 ;a;1;c1;1 = a;_1,.#c1.1 = a;_1,.#c.a = ;a;a;c;1 2a) ;a;b1;c;1 = a;_1,.#c.b1 = (a;_1,.#c.1);_1,.#c.b = ;(;a;1;c;1);b;c;1 2b) = (a;_1,.#c.b);_1,.#c.b = ;(;a;b;c;1);b;c;1 ;a;1;1;11 = a;_2,1 = a;_1,.#a.a = ;a;a;a;1 ;a;1;1;d1 = a;_d1,1 = a;_d,.#a.a = ;a;a;a;d ;a;1;c1;d = a;_d,.#c1.1 = a;_d,.#c.a = ;a;a;c;d 2a) ;a;b1;c;d = a;_d,.#c.b1 = (a;_d,.#c.1);_d,.#c.b = ;(;a;1;c;d);b;c;d 2b) = (a;_d,.#c.b);_d,.#c.b = ;(;a;b;c;d);b;c;d ;a;1;1;1;1 = a;_1,1,1 = a;_a,.#a.a = ;a;a;a;a ;a;1;1;1;e1 = a;_1,e1,1 = a;_a,e,.#a.a = ;a;a;a;a;e ;a;1;1;d1;e = a;_d1,e,1 = a;_d,e,.#a.a = ;a;a;a;d;e ;a;1;c1;d;e = a;_d,e,.#c1.1 = a;_d,e,.#c.a = ;a;a;c;d;e 2a) ;a;b1;c;d;e = a;_d,e,.#c.b1 = (a;_d,e,.#c.1);_d,e,.#c.b = ;(;a;1;c;d;e);b;c;d;e 2b) = (a;_d,e,.#c.b);_d,e,.#c.b = ;(;a;b;c;d;e);b;c;d;e

;a;1;...;1 [1#k,k>1] = ;a;...;a [a#k] ;a;1;...;1;m1;Q [1#k,k1>1] = ;a;...;a;m;Q [a#k1] 2a) ;a;b1;Q = ;(;a;1;Q);b;Q 2b) = ;(;a;b;Q);b;Q

(1b ;A;;1 = ;A;...;A [a#a] ;A;;11 = ;(;A;;1);;1 2a) ;A;;b1 = ;(;A;;1);;b 2b) = ;(;A;;b);;b 2a1) ;A;;B;;c1 = ;A;;(;B;;1);;c 2a2) = ;(;A;;1);;(;B;;1);;c 2b2) = ;(;A;;B;;c);;(;A;;B;;c);;c

;A;.#r1.1 = ;A;.#r....;.#r.A [A#A]

2a) ;A;.#r.b1 = ;(;A;.#r.1);.#r.b 2b) = ;(;A;.#r.b);.#r.b

2a1) ;A;.#r....;.#r.M;.#r.z1 = ;A;.#r....;.#r.(;M;.#r.1);.#r.z 2a2) = ;(;A;.#r.1);.#r....;.#r.(;M;.#r.1);.#r.z P=(;A;.#r....;.#r.M;.#r.z) 2b2) = ;P;.#r....;.#r.P;.#r.z

;;1;a = ;a;.#a.a ;;f1;a = ;;f;a;.#a.a ;;.#s.O;a;.#rP = ;;.#s.O;(;a;.#rP)

3a) ;;f;;1;a = ;;(;;f;a);a (3b = ;;(;;f;a);a;.#a.a 4a) ;;g1;;f1;a = ;;(;;g1;;f;a);a;.#a.a 4b) = ;;g;;(;;g1;;f;a);a;.#a.a

4a) ;;j;;...;;g;;f;;1;a = ;;(;;j;;...;;f;a);a;.#a.a 4b) = ;;j;;...;;g;;(;;j;;...;;f;a);a;.#a.a 4a) ;;j;;...;;g1;;f1;a = ;;(;;j;;...;;g1;;f;a);a;.#a.a 4b) = ;;j;;...;;g;;(;;j;;...;;g1;;f;a);a;.#a.a

;;F;;;1;a = ;;(;;F;a);;...;;(;;F;a);a;.#a.a [(;;F;a)#(;;F;a)] ;;G;;;f1;a = ;;(;;G;;;f;a);;...;;(;;G;;;f;a);a;.#a.a [(;;G;;;f;a)#(;;G;;;f;a)] ;;F;.#r1.1;a = ;;(;;F;a);.#r....;.#r.(;;F;a);a;.#a.a [(;;F;a)#(;;F;a),r>1] ;;G;.#r1.f1;a = ;;(;;G;.#r1.f;a);.#r....;.#r.(;;G;.#r1.f;a);a;.#a.a [(;;G;.#r1.f;a)#(;;G;.#r1.f;a),r>1]

;.#r1.1;a = ;.#r.(;a);.#r....;.#r.(;a);a;.#a.a [(;a)#(;a),r>1] ;.#r1.f1;a = ;.#r.(;.#r1.f;a);.#r....;.#r.(;.#r1.f;a);a;.#a.a [(;.#r1.f;a)#(;.#r1.f;a),r>1] ;.#s.F;.#r1.1;a = ;.#s.(;.#s.F;a);.#r....;.#r.(;.#s.F;a);a;.#a.a [(;.#s.F;a)#(;.#s.F;a),s>1,r>1] ;.#s.G;.#r1.f1;a = ;.#s.(;.#s.G;.#r1.f;a);.#r....;.#r.(;.#s.G;.#r1.f;a);a;.#a.a [(;.#s.G;.#r1.f;a)#(;.#s.G;.#r1.f;a),s>1,r>1]

 :1 = 1
 :11 = ;
 :111 = (
 :1111 = )
 :11111 = :(X)
 :111111 = (X)#(N)
 :1111111 = -
 :11111111 = ω(N)
 :111111111 = 0
 :1.#n.
Hel10 wwwØrld,

with the introduction of a binary notation for characters we can now have a countable number of types of operator-separators ;_a,... ('soeparators') and use these to further expand the earlier algorithms for creating big numbers. Each algorithm can function in its own universe U_a as well as in all higher algorithm universes U_n>a

Looking back at how we came here: We started by counting units 1... in U
We introduced the operator ; in U₁ with x;1=x+x and the next operations in its first row. In the second row of U₁ we defined ;... as countable, which made further rows and higher dimensions of U₁ possible.
Then came the need for a new type of operator, but instead we defined an operator function U₂ starting at ;x;y;z and a subsequent first row of single separators ; representing expressions of U₁ with virtual types of operators ;_d,e,...
With the second row of the parameter array of U₂ the separators ;... became countable, and similarly to the definition of dimensions in U₁ we built up U₂ as a multi-dimensional parameter array.
U₃ started with ;;f;x which renders operator function types countable by a 'negative side' which counts the function types. But instead of evaluating U₃ as if it was a continuation of U₂ with virtual countable types of separators ;_n positioned at its far end, we let each evaluation step that reduces U₃ feedback U₂ and (in version 3b) increase U₂ by itself.

All expressions of U_a should (principally at least) still be reducable, and U₃ is. Having multiple lower universe numbers feedback and increase while the expression in the highest universe is reduced seems to produce the biggest numbers, and is the way to push this thing forward.
Cantor's idea was to postulate infinity and define it as a number that is bigger than any counted number. He started counting from there and then he counted infinities and infinite infinities and so on. But Cantor's ω is nothing more than a mathematical character abiding by some algorithmic rules, which all can be made countable. On top of that many further types of infinities can be postulated too.
Not much later the concept that types infinity as a monolithic realm may prove to be long overdue and be replaced by something more general, pervading the whole of number space, increasing at all relevant number sites at once.

Eventually, just as the algorithms up to U₃ describe all possible variations of 1; totally, the binary notation by countable characters 1: will become complete. This is where the algorithms to form and count formal languages, meta-languages included, cannot be expanded into higher universes.
The informal language of philosophy may take us further from there, but there is no chance we can formalize (know for sure) that it does. As the philosophers' heavens transcend counting, if a seventh heaven exists, it must be the produce of formal language.

Franzes van NovaLoka
the Hague, 18-10-2008

Biggest "Book of Records Number" the day before yesterday in BBNS-1b-2b2-3b-4b: