Well ordering principles and $Π^1_4$-statements: a pilot study

In previous work, the author has shown that $Π^1_1$-induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of J.-Y. Girard's dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called $2$-ptykes. We show that $Π^1_2$-induction along $\mathbb N$ is equivalent to the existence of fixed points for all $2$-ptykes that satisfy a certain normality condition. Beyond this specific result, the paper paves the way for the analysis of further $Π^1_4$-statements in terms of well ordering principles.

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