Order of Parentheses is Irrelevant: Metatheorem?

Question

Here was shown by induction that the order of parentheses is irrelevant when associativity is verified.

Question: Would this be a metatheorem about the formal language (say, of ZF) where the metalanguage is English and where the Induction Principle used is only an intuitive (informal) one that we accept on purely logical/intuitive grounds?

Answer 1

Yes.

Consider for example propositional logic; in the calculus you can derive e.g. :

$\vdash ((p_1 \lor p_2) \lor p_3) \leftrightarrow (p_1 \lor (p_2 \lor p_3))$

but there is no way to derive in the calculus a "one shot" formula expressing the fact that associativity holds with $n$ whatever.

To prove it, we have to use induction in the meta-theory.