Proving incompleteness without unbound quantifiers

Question

I am reading An Introduction to Godel's theorems. On page 113, the author aims to prove that the language of Peano arithmetic can express every primitive recursive function, and later proves that it can also 'capture' every primitive recursive function. The plan for that is as follows:

But the language of PA only has the successor function, which is a basic function. It has the functions of addition and multiplication, which are both primitive recursive. Every atomic wff is enumerable in one step, since the only predicate is identity which is expressible through the basic function of projection. The logical connectives all preserve primitive recursion too. The only thing that doesn't preserve primitive recursion in the language of PA is the unbounded quantifiers. If we weakened the language of PA to only have bounded quantifiers, wouldn't it be the case that the language can express only primitive recursive functions? Would it still be possible to prove incompleteness from it?

Answer 1

In fact, $\Delta^0_0$ (= expressible using only bounded quantifiers) captures a proper subclass of the p.r. functions; see here. Meanwhile the set of $\Delta^0_0$ theorems of $\mathsf{PA}$ is decidable and coincides with the set of true $\Delta^0_0$ sentences - this is a consequence of the $\Sigma_1$-completeness of $\mathsf{PA}$ - so Godel does not apply to this fragment.