Abstract

It is well-known (due to C. Parsons) that the extension of primitive recursive arithmeticPRA by first-order predicate logic and the rule ofΠ 2 0 -inductionΠ 2 0 -IR isΠ 2 0 -conservative overPRA. We show that this is no longer true in the presence of function quantifiers and quantifier-free choice for numbersAC 0,0-qf. More precisely we show that ℐ :=PRA 2 +Π 2 0 -IR+AC 0,0-qf proves the totality of the Ackermann function, wherePRA 2 is the extension ofPRA by number and function quantifiers andΠ 2 0 -IR may contain function parameters. This is true even forPRA 2 +∑ 1 0 -IR+Π 2 0 -IR −+AC 0,0-qf, whereΠ 2 0 -IR − is the restriction ofΠ 2 0 -IR without function parameters.