User blog:Ytosk/Transfinite Extensions of BMS

Bubby3 has made a Transfinite BMS (TBMS), but i couldn't find the full definition, so i will just link his blog posts about it: https://googology.wikia.org/wiki/User_blog:Bubby3/Transfinite_BMS. https://googology.wikia.org/wiki/User_blog:Bubby3/Transfinite_BMS_extension

But he wasn't the only one who thought of that. I made a different extension, not knowing about his extensions. Also, on Bubby3's blog post, Alemagno12 created another different idea. In this blog post, i will explain Alemagno12's extension and my extension.

Alemagno12's extension: We can rewrite the columns using matrices. Adding a (0) at the end means adding 1 to the first entry in the bracket. Examples: ((0))=(1), ((0)(0))=(2), ((0)(0)(0)(0))=(4) When an entry becomes infinite, we change it back to 0 and increase the next entry by 1: ((0)(1))=(0,1), ((0)(1)(0)(1)(0)(0))=(2,2), ((0)(1)(1))=(0,0,1), ((0)(1)(1)(1))=(0,0,0,1) Using only (0) and (1), we can describe all the finite matrices. Before we continue to transfinite, i will change the previous rule so that it doesn't change the infinite entry to 0, but to 1, because it's more convenient: ((0)(1)(1))=(1,1,1), instead of (0,0,1). The first transfinite matrix is ((0)(1)(2))=(1,1,1,1...). Then we can have ((0)(1)(2)(3)), ((0)(1,1)), ((0)(1,1,1)), ((0)((0)(1)(2))), ((0)((0)((0)(1)(2))))... and the limit of this is ((0)((0)((0)((0)(...))))).

My extension: From now on, i will not write any of the zeroes at the end of brackets. I wasn't as creative when started making this, so i just defined the limits or the undefined things when i got to them: (1,1,1,...)=() Now, we have infinitely many ones. We can add other things before them: ()(2,)[2]=()(2,1,1,1)[4]=()(2,1,1)(3,2,2,)... ()(2,2,)[2]=()(2,2,1,1)[4]=... To get an infinite amount of a different number, we put the number before. We can then put things after : (,1)[2]=()(2)(3)(4)[4]=()(2)(3)(4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4)[16]... Again, infintely many ones becomes : ()[2]=(,1,1,1,1)[4] We, once again, have multiple brackets, which means the rest can work like bms: ((1))=(...) ((1,1))=((1)(2)(3)...) (())=((1,1,1...)) The limit of this=((((...)))) As i said, i just defined the limits when i got to them. At first, that resulted in a lot of chaos, but then i compressed it without losing any of its power: We can have subscripts of brackets. It works similarly to backslashes in BAN. (_2 1)=(((...))) (_2 2)=(((...)_2 1)_2 1) (_2 1,1)[n]=(_2 f(n))[f(n)] (_2)=(_2 1,1,1...) (_2(_2 1))=(_2(((...)))) (_2_2 1)=(_2(_2(_2(...)))) (_2_2 1)=(_2_2_2...) (_2_2_2 1)=(_2_2_2_2...) (_2(1)_2 1)=(_2_2_2_2...1) (_2(1)_2(1)_2 1)=(_2(1)_2_2(1)_2...1) (_2(1)_2(2)_2 1)=(_2(1)_2(1)_2(1)_2...1) (_2(1,1)_2 1)=(_2(1)_2(2)_2...1) ... (_2()_2 1)=(_2(1,1,1...)_2 1) (_2((_2 1))_2 1)=(_2((((...))))_2 1) (_2(_2 1)_2 1)=(_2((_2((...))_2 1))_2 1) Just like _2 changes into a copy of the innermost containing the _2, _3 changes into a copy of the innermost _2 containing the _3. It works similarly with _4 etc. (_2(_2_3 1)_2 1)=(_2(_2(_2(...)_2 1)_2 1)_2 1) (_2(_2_3 2)_2 1)=(_2(_2(_2(...)_2 1_2_3 1)_2 1_2_3 1)_2 1) (_2(_2_3_2 1)_2 1)=(_2(_2_3(_2(_2_3(...))_2 1))_2 1) (_2(_2_3_2 1)_2 1)=(_2(_2_3_2_3...)_2 1) (_2(_2_3_2_3 1)_2 1)=(_2(_2_3_2(_2_3_2(...)_2 1)_2 1)_2 1) (_2(_2_3(1)_2 1)_2 1)=(_2(_2_3_2_3_2_3... 1)_2 1) (_2(_2_3(1)_2_3 1)_2 1)=(_2(_2_3(1)_2(2,_2_3(1)_2(...)_2 1)_2 1)_2 1) (_2(_2_3(1,1)_2 1)_2 1)=(_2(_2_3(1)_2_3(2)_2_3... 1)_2 1) (_2(_2_3_3 1)_2 1)=(_2(_2_3(_2_3(...)_2 1)_2 1)_2 1) (_2(_2_3(1)_3 1)_2 1)=(_2(_2_3_3_3...1)_2 1) (_2(_2_3(_2_3_4 1)_3 1)_2 1)=(_2(_2_3(_2_3(...)_3 1)_3 1)_2 1) (_{1,1}1)=(_2(_2_3(_2_3_4(...)_4 1)_3 1)_2 1) etc.

Before making this post, i decided to compare Alemagno12's extension with my extension, and until (0)((0)(1,1))=((1,1)), the translation into my extension was removing the (0)'s and (1)'s and subtracting 2 from everything. Example: (0)((0)(1)(2)(3)(4)(5)(5)(3)(4)(5)(4)(2))=((1)(2)(3)(3)(1)(2)(3)(2)) After ((1,1)), they are almost the same, except my extension has the subscripts, so it goes beyond (_2 1)