A note on the $\Pi^0_2$ -induction rule

Abstract. It is well-known (due to C. Parsons) that the extension of primitive recursive arithmetic PRA by first-order predicate logic and the rule of $\Pi^0_2$-induction $\Pi^0_2$ -IR is $\Pi^0_2$-conservative over PRA. We show that this is no longer true in the presence of function quantifiers and quantifier-free choice for numbers AC $^{0,0}$-qf. More precisely we show that ${\cal T} :=$PRA $^2 +\Pi^0_2$-IR $+$AC $^{0,0}$-qf proves the totality of the Ackermann function, where PRA $^2$ is the extension of PRA by number and function quantifiers and $\Pi^0_2$-IR may contain function parameters. This is true even for PRA $^2 +\Sigma^0_1$-IR $+\Pi^0_2$ -IR $^- +$AC $^{0,0}$-qf, where $\Pi^0_2$-IR $^-$ is the restriction of $\Pi^0_2$-IR without function parameters.