## FANDOM

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Recently, Taranovsky has updated his ordinal notation page. In the last section he defined another ordinal notation system. In that system he used a ternary function C, and $$C_i(a,b)$$ corresponds to $$C(\Omega_2\times i+a,b)$$ in the main system.

Taranovsky conjectured that using $$i\le n$$ (where n is a positive integer), the last system reaches $$\Pi_n^1\text{-TR}_0$$. As a result, in the main system, $$C(C(\Omega_2\omega,0),0)$$ reaches second-order arithmetic.

I've done some comparisons between array notation and Taranovsky's ordinal notation (but only a part of them published on my site). If the conjecture and the comparisons work, s(n,n{1{1(2{2+}2)+2}2(2{2+}2)+2}2) will surpass all functions provably recursive in second-order arithmetic.