User blog comment:Hyp cos/Suggestions about Size Classes of Numbers/@comment-3427444-20170701135427/@comment-27513631-20170705201444

n-ple recursion (n-ary analogue of double recursion), added to PRA, should have strength \(\omega^{\omega^n}\). PA with just \(\Pi^0_n\) induction has PTO \(\omega\uparrow\uparrow (n+1)\) (that is, \(\Pi^0_1 \Rightarrow \omega^\omega\), \(\Pi^0_1 \Rightarrow \omega^{\omega^\omega}\), etc.). Unfortunately, the former was obtained through intuitive reasoning, and I can't find the source for the latter.