Power Diamond

Best to read Little Omega

Power diamond!

b(n) = n↑ⁿn↑ⁿn↑ⁿn…(n ↑ⁿs)…↑ⁿn

b¹(n) = b(n) ↑^b(n)b(n) ↑^b(n)…(b(n) “↑^b(n)”s)…b(n)

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b^ω(n) = bⁿ(n) ↑^{b^n(n)} bⁿ(n) ↑^{b^n(n)} bⁿ(n) ↑^{b^n(n)}…(bⁿ(n) “↑^{b^n(n)}”s)…bⁿ(n)

π¹(n) = b^ω(b^ω(b^ω(b^ω(b^ω(b^ω(…(n “b^ω(“s)…b^ω(n)

π²(n) = π¹(π¹(π¹(π¹(π¹(π¹(π¹(… (π¹(n) “π¹(“s) π¹(n)

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π^ω(n) = πⁿ(πⁿ(πⁿ(… (πⁿ(n) “πⁿ(“s ) πⁿ(n)

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π^ε0(n) = π^{ω+n^n}(π^{ω+n^n}(π^{ω+n^n}(…(π^{ω+n^n}(n) “π^{ω+n^n}(“s)…π^{ω+n^n}(n)

M(n,o) = π^ε0(ω(n,o,n,o,n)) = n◊o

n◊o◊p = M(M(…(p “M(“s)…M(n,o),M(n,o)…) (we’ll call this A for simplicity)

2◊2◊2 = M(M(2,2),M(2,2)) (we’ll call this B for simplicity)

n◊o◊p◊q = M(…(q “M(“s)…M(a,a),M(a,a)…)

2◊2◊2◊2 = M(M(B,B),M(B,B))

And so on with the recursion.

Note: if you want to write n◊n◊n…(x ◊s)…◊n , it can be simplified as n◊^xn.