User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-11227630-20180208013330/@comment-1605058-20180209101728

2 and 3 cannot be expressed in FOST. To make sense of those we would need to expand the language by additional symbols, e.g. for 3 it would be a function symbol together with axiom schema stating that for finite n, f(n) is \(\Sigma_n\)-correct. Further, the KP axiom schemata should be extended to include formulas using that symbol.

With that, I believe we have 1 > 2 and 3 > 4 strength-wise. For 2 vs 3, I think that depends on details. If we add schema adding for each n and \(\varphi\) \(\Sigma_n\) the formula "f(n) is correct about \(\varphi\)", I think 2 > 3, but if we add "for n > m, f(n) is correct about \(\varphi\)", then 2 = 3.