User blog:Bubby3/Transfinite BMS extension

A discord post that I posted says this. "We can extend further by using (1(2,0)1) as equal to (1[1]1) = (1,,1), (1(2,0)(2,0)1) = (1,,,1), (1(2,0)(3,0)1) = the limit of TBMS. We can continue to (1(2,1)1) = (2,1) row matrix, which (0)(1(2,1)1) is equal to (0)(1(2)(3...(n,1)...3)1), or the limit of (0)(1,1), (0)(1(2)(3,1)1), (0)(1(2)(3(4)(5,1)1)1), etc. Next up is (1(2,1)(3,2)1), then (1(2,1,1)1). Then (1(2,2)1) is the limit of (1,1), (1(2,1)1), (1(2(3,1)1)1), etc. Then there is (0)(1(2,2,2)1), then (0)(1(2(3,3)2)1). Limit of this system is (0)(1(2....(n)...2)1).". Here is a summary of this. After (1[n]1), BMS can be extended to transfinite arrays. (1[n]1) = (1(2)...(2)1) with n (2)'s. This continues like normal BMS, so here are the expressions
 * (1(2,1)1) is the limit of (1(2)(3(4)(5...1)1)1)
 * (1(2,1,1)1) is the limit of (1(2,1)(3,2(4,2)(5,3...1)1)1)
 * (1(2,2)1) is the limit of (1(2(3(...)1)1)1)
 * (1(2,2,1)1) is the limit of (1(2,2)(3,3(4,4)(5,5(...)1)1)1)

Limit is (0)(1(2(3(4(5...5)4)3)2)1)

We can go further by adding slashes. A * means that entry is until the slash Now // controls / like / controls /. We can now have mutliples slashes like (1///1), (1////1), etc. Now, /(n) means transfinitely many slashes. \ does the same but with number of slashes. Adding a slash means fixed point of the order, and backslash means the number of slashes.
 * (0)(1/1) = (0)(1(2(3(4(5...5)4)3)2)1)
 * (0)(1/1)(1/1) = (0)(1/1)(1(2*/1)(2(3*/1)2)1)
 * (0)(1/1)(2/1) = (0)(1/1)(2(3,2*/1)(4(...)2)1)
 * (0)(1/1)(2/2) = (0)(1/1)(2(3)2).
 * (0)(1/1,1) = (0)(1/1)(2/2)(3/3)(4/4)...
 * (0)(1/1/1) = (0)(1/1(2/2(3/3(...)3)2)1)
 * (0)(1/(2)1) = (0)(1/1/1...1/1/1)

In my array notation. (1(a)/(b)\(c)1) can be converted to (1[a`b`c]1). {} represent dimensional seperator. The limit of this notation is is (1[0{0{...0`1...}1}1]1)