There is of course a difference for logicians, but from a non-logician-mathematician's perspective, what is the real significance of arbitrarily complex induction predicates?
- Are there perhaps examples from ''ordinary'' mathematics where ridiculously (or even arbitrarily) high arithmetical quantifier complexities are implicitly involved?
Comment: I am aware that in the definition of the analytic hierarchy, the $\Delta^1_0$-predicates are defined so as to encompass all of the arithmetical predicates, and so one might be tempted to argue that, in analysis, we are constantly quantifying over sets defined by formulas with arbitrarily high arithmetic quantifier complexities. But again, is this actually tracking something in mathematics outside of logic? Or could we just as well do with analytical predicates with arithmetical quantifiers restricted to some fixed bound?
- Concretely, what more are we assuming when we go from I$\Sigma_n$ to I$\Sigma_{n+1}$?
Comment: I'm somewhat tempted to say that we are implicitly assuming more sets of numbers to exist/be well-defined, especially in view of the difference between SSy$(M)$ and SSy$(M')$ for non-standard models $M\models$I$\Sigma_n \land \neg$ I$\Sigma_{n+1}$ and $M'\models$I$\Sigma_{n+1}$ (comp. Theorem 2.7 here: https://bpb-us-w2.wpmucdn.com/blog.nus.edu.sg/dist/4/10956/files/2022/06/3_wkl.pdf) . But then, one can hardly say that assuming stronger induction simply amounts to stronger set existence assumptions, in view of $PA$'s equivalence with $ZFC$ with the negated infinity axiom, explicitly excluding infinite sets. Perhaps, in view of the correspondence of induction and separation (to use Wong's words, see the comment before Thm 2.7), one could say that stronger induction corresponds to stronger "relative set existence assumptions" (as separation is stating the existence of definable subsets relative to an already given set)? But all this still seems muddled to me.
- Same questions for the analytical hierarchy: what is the mathematical significance of $\Pi^1_{27}$? $\Pi^1_{27000}$?
Edit: Another thought related to the comment to question 1.: in view of the existence of universal $\Pi^1_1$-sets, all arithmetical sets are many-one-reducible to one set defined by a fixed number of arithmetical quantifiers and just one analytical quantifier. Perhaps this is an instance where in analysis, we employ techniques that are (implicitly) making exhaustive use of the entire arithmetical hierarchy? But again, what are the applications outside of logic?