When I can assign to a statement (of PA) a classification $\Sigma_n^0$ (resp. $\Pi_n^0$) of the arithmetical hierarchy (AH), then this also classifies it as $\Sigma_m^0$ (resp. $\Pi_m^0$) for any $m\ge n$. So the only interesting question is how far down it can be placed in this hierarchy. A bit of thinking about this brought me to the following realization:
I always thought that math was about proving theorems (with theorems formally written down as statements $\varphi$). But might it be that math is more generally about pushing as many statements as far down in AH as possible?
As soon as a statement is proven, we showed that it is a tautology, that it is equivalent to some statement with bounded quantifiers, e.g. $1=1$.
- Is really any provable statement already in $\Sigma_0^0$ or $\Pi_0^0$?
- If so, what is the point of placing statements in AH? Is it about classifying unprovable (independent or wrong) statements?
- If so, what is the point of categorize unporvable statements? E.g. I know that a statement in $\Pi_1^0$ is already shown to be true in the standard natural numbers $\Bbb N$ if we can show its independence from PA. So it is not provable, but still "sufficiently true" from our meta point of view. But is there more about this? What does it help to categorize wrong statements? Does it help us to place $\varphi$ in AH during the process of proving it?
I read that a statement is in $\Sigma_0^0$ and $\Pi_0^0$ if it is logically equivalent to a statement with only bounded quantifiers. Does this "logically" mean that in order to show the equivalence, I am only allowed to use the axioms and rules of the deductive system I am working in but not the axioms of PA? In this case my question indeed would be very trivial because based on a false understanding.