I’ve been trying to get my head around the significance of (i) the relation between ACA0 and ACA (the theory you get by keeping arithmetic comprehension, but allowing induction for any second-order wff) and (ii) the relation between ACA and T(PA) (the theory you get by adding to first-order PA a Tarski-like truth-theory AND allowing the new truth-predicate to appear in induction axioms). The technicalities hereabouts are reasonably clear: but — as I say — their philosophical significance is not. So watch this space.
We might be tempted by the following argument for being generous with induction in moving from (i) ACA0 to (ii) ACA, or indeed in moving from (i’) PA + T, the result of taking PA and bolting on a truth-theory “from outside” while leaving the induction axioms untouched, to (ii) T(PA):
Instances of induction are conditionals, telling us that from F(0) and (Ax)(Fx -> Fx’) we can infer (Ax)Fx. So we can derive(Ax)Fx. when we have already established the corresponding premiss (i) F(0) and can also establish (ii) (Ax)(Fx -> Fx’). But if we can already establish (i) and (ii) then trivially we can (iii) case by case derive each and every one of $F(0), F(1), F(2), … However, there are no ‘stray’ numbers which aren’t denoted by some numeral; so that means (iv) that we can show of each and every number that F holds of it. What more can it possibly take for F to express a genuine property that indeed holds for every number, so that (v) (Ax)Fx is true? In sum, it seems that we can’t possibly overshoot by allowing instances of the Induction Schema for any open wff of the language we are working with. The only usable instances from our generous range of axioms will be those where we can in fact establish the antecedents (i) and (ii) of the relevant conditionals — and in those cases, we can’t go wrong in accepting the consequents (v) too.
This argument for inflating ACA0 to ACA, or PA + T to T(PA) — and thereby getting to be able to prove new arithmetical sentences like Con(PA) — is surely too easy! But saying exactly why isn’t so easy …